# Informationally Complete Characters for Quark and Lepton Mixings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Minimal Informationally Complete Quantum Measurements

## 2. Informationally Complete Characters for Quark/Lepton Mixing Matrices

#### 2.1. Groups in the Series $\Delta (6{n}^{2})$ and More Groups

#### 2.2. Exceptional Subgroups of $SU(3)$

## 3. Generalized CP Symmetry, CP Violation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Small groups considered in our Table 3. For each group and each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations. According to our demands, each selected group has both 2- and 3-dimensional characters (with at least one of them faithful) that are magic states for an informationally complete POVM (or MIC), with the rank of Gram matrix equal to ${d}^{2}$. The Pauli group performing this action is in general a d-dit but is a 2-qutrit (2QT) for the group $(120,5)=SL(2,5)=2I$, a 3-qutrit (2QT) for the group $(360,51)={\mathbb{Z}}_{3}\times SL(2,5)$ or may be a three-qubit/qutrit (3QB-QT) for the groups $(648,532)$ and $(648,533)$.

Group | d | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(24,12) | 5 | 1 | 1 | 2 | 3 | 3 | ||||||||||

5-dit | 5 | 21 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||

(120,5) | 9 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | ||||||

9-dit | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 79 | ${d}^{2}$ | |||||||

2QT | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||

(150,5) | 13 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | ||

13-dit | 13 | 157 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||

(72,42) | 15 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

15-dit | 15 | 203 | 209 | 209 | 195 | 195 | 219 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

(216,95) | 19 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

19-dit | 19 | 343 | 357 | 359 | 355 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 6 | 6 | 6 | |||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||||

(294,7) | 20 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

20-dit | 20 | 349 | 388 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 6 | ||||||||||||

390 | 390 | 390 | 398 | 398 | ||||||||||||

(72,3) | 21 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

21-dit | 21 | 405 | 405 | 421 | 421 | 421 | 421 | 421 | 421 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

2 | 2 | 2 | 3 | 3 | 3 | |||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||

(162,12) | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

22-dit | 22 | 446 | 463 | 463 | 463 | 463 | 473 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | ||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 198 | ||||||||||

(162,14) | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

22-dit | 22 | 444 | 461 | 463 | 461 | 463 | 473 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | ||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 198 | ||||||||||

(648,532) | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

24-dit | 24 | 527 | 527 | 562 | ${d}^{2}$ | ${d}^{2}$ | 560 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3QB-QT | 24 | 500 | 500 | 476 | 568 | 568 | 448 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||||||||

24-dit | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 564 | ${d}^{2}$ | ${d}^{2}$ | 552 | 552 | |||||||

3QB-QT | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 448 | 560 | 560 | 510 | 510 | |||||||

(648,533) | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

24-dit | 24 | 539 | 539 | 562 | ${d}^{2}$ | ${d}^{2}$ | 514 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 574 | 574 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3QB-QT | 24 | 532 | 532 | 481 | 572 | 572 | 452 | 572 | 568 | 568 | 570 | 570 | 572 | 575 | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||||||||

24-dit | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 563 | ${d}^{2}$ | ${d}^{2}$ | 478 | 478 | |||||||

3QB-QT | ${d}^{2}$ | 573 | 573 | 575 | 488 | 560 | 560 | 520 | 520 |

**Table A2.**The following up of Table A1.

Group | d | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(120,37) | 25 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |

25-dit | 25 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 623 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||

(360,51) | 27 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

3QT | 27 | 613 | 613 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | |||||

727 | 725 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | |||||

(162,44) | 30 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

30-dit | 31 | 826 | 861 | 871 | 861 | 871 | 883 | 877 | 879 | 883 | 898 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 898 | |

2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||

898 | 898 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

(600,179) | 33 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

33-dit | 33 | 1041 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

6 | 6 | 6 | ||||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||||||||

(168,45) | 35 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

35-dit | 35 | 1175 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | ${d}^{2}$ | |

2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

3 | 3 | 3 | 3 | 3 | ||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||||||

(480,221) | 36 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 |

36-dit | 36 | 36 | 1085 | 1185 | 1184 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1278 | 1278 | 1278 | |

3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | ||

1278 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1275 | 1278 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1277 | 1273 | ||

5 | 5 | 6 | 6 | 6 | 6 | |||||||||||

1294 | 1294 | 1295 | 1295 | 1295 | 1295 |

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**Figure 1.**An angular picture of the three generations of quarks and leptons. The blue and black pancakes have isospin $1/2$ and $-1/2$, respectively. The inner and outer rings have weak hypercharges $\frac{1}{3}$ and $-1$, respectively.

**Table 1.**(1) The three generations of up-type quarks (up, charm and top) and of down-type quarks (down, strange and bottom) and, (2) the three generations of leptons (electron, muon and tau) and their partner neutrinos. The symbols Q, ${T}_{3}$ and ${Y}_{W}$ are for charge, isospin and weak hypercharge, respectively. They satisfy the equation $Q={T}_{3}+\frac{1}{2}{Y}_{W}$.

Matter | Type 1 | Type 2 | Type 3 | Q | ${\mathit{T}}_{3}$ | ${\mathit{Y}}_{\mathit{W}}$ |
---|---|---|---|---|---|---|

(1) quarks | u | c | t | 2/3 | 1/2 | 1/3 |

d | s | b | −1/3 | −1/2 | 1/3 | |

(2) leptons | e | $\mu $ | $\tau $ | −1 | −1/2 | −1 |

${\nu}_{e}$ | ${\nu}_{\mu}$ | ${\nu}_{\tau}$ | 0 | 1/2 | −1 |

**Table 2.**Experimental values of the angles in degrees for mixing patterns of quarks (in the CKM matrix) and leptons (in the PMNS matrix).

Angles (in Degrees) | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{13}$ | ${\mathit{\theta}}_{23}$ | ${\mathit{\delta}}_{\mathit{CP}}$ |
---|---|---|---|---|

quark mixings | 13.04 | 0.201 | 2.38 | 71 |

lepton mixings | 33.62 | 8.54 | 47.2 | −90 |

**Table 3.**List of the $16+2$ groups with the number of conjugacy classes $cc\le 36$ that satisfy rules (a) and (b). As mentioned in Section 3, groups $(294,7)$ and $(384,568)$ need two CP phases to become viable models. The smallest permutation representation on $k\times l$ letters stabilizes the n-partite graph ${K}_{k}^{l}$ given in the fourth column. The group $\Delta (6\times {n}^{2})$ is isomorphic to ${\mathbb{Z}}_{n}^{2}\u22ca{S}_{3}$. A reference is given in the last column if a viable model for quark and/or lepton mixings can be obtained. The extra cases with reference † and ‡ can be found in [18,21], respectively.

Group | Name or Signature | $\mathit{cc}$ | Graph | Ref. |
---|---|---|---|---|

SmallGroup(24,12) | ${S}_{4}$, $\Delta (6\times {2}^{2})$ | 5 | ${K}_{4}$ | [15] |

SmallGroup(120,5) | 2I, SL(2, 5) | 9 | ${K}_{5}^{3}$ | [20] †,‡ |

SmallGroup(150,5) | $\Delta (6\times {5}^{2})$ | 13 | ${K}_{5}^{3}$ | [2,15,16] |

SmallGroup(72,42) | ${\mathbb{Z}}_{4}\times {S}_{4}$ | 15 | ${K}_{3}^{4}$ | [9] |

SmallGroup(216,95) | $\Delta (6\times {6}^{2})$ | 19 | ${K}_{6}^{3}$ | [15] |

SmallGroup(294,7) | $\Delta (6\times {7}^{2})$ | 20 | ? | [31] |

SmallGroup(72,3) | ${Q}_{8}\u22ca{\mathbb{Z}}_{9}$ | 21 | ${K}_{2}^{3}$ | [9] |

SmallGroup(162,12) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$ | 22 | ${K}_{9}^{3}$ | [2,15,18] |

SmallGroup(162,14) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$, ${D}_{9,3}^{(1)}$ | 22 | ${K}_{9}^{3}$ | [2,15,19] |

SmallGroup(384,568) | $\Delta (6\times {8}^{2})$ | 24 | ? | [31] |

SmallGroup(648,532) | $\Sigma (216\times 3)$, ${\mathbb{Z}}_{3}\u22ca({\mathbb{Z}}_{3}\u22caSL(2,3))$ | 24 | ? | [15,22] |

SmallGroup(648,533) | Q(648), ${\mathbb{Z}}_{3}\u22ca({\mathbb{Z}}_{3}\u22caSL(2,3))$ | 24 | ? | [15,17] |

SmallGroup(120,37) | ${\mathbb{Z}}_{5}\times {S}_{4}$ | 25 | ${K}_{5}^{4}$ | † |

SmallGroup(360,51) | ${\mathbb{Z}}_{3}\times SL(2,5)$ | 27 | ${K}_{12}^{6}$ | † |

SmallGroup(162,44) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$ | 30 | ${K}_{9}^{3}$ | [15] |

SmallGroup(600,179) | $\Delta (6\times {10}^{2})$ | 33 | ${K}_{10}^{3}$ | [2,15,16] |

SmallGroup(168,45) | ${\mathbb{Z}}_{7}\times {S}_{4}$ | 35 | ${K}_{7}^{4}$ | † |

SmallGroup(480,221) | ${\mathbb{Z}}_{8}.{A}_{5}$, $SL(2,5).{\mathbb{Z}}_{4}$ | 36 | ${K}_{8}^{6}$ | ‡ |

**Table 4.**For each of the first three small groups considered in our Table 3 and the group $(294,7)$ added in Section 3, for each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations. According to our requirement, each selected group has both 2- and 3-dimensional characters (with at least one of them faithful) that are fiducial states for an informationally complete POVM (or MIC) with the rank of Gram matrix equal to ${d}^{2}$. The Pauli group performing this action is a d-dit or a 2-qutrit (2QT) for the group $(120,5)=SL(2,5)=2I$.

Group | d | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(24,12) | 5 | 1 | 1 | 2 | 3 | 3 | ||||||||

5-dit | 5 | 21 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||

(120,5) | 9 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | ||||

9-dit | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 79 | ${d}^{2}$ | |||||

2QT | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||

(150,5) | 13 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

13-dit | 13 | 157 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

(294,7) | 20 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

20-dit | 20 | 349 | 388 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 6 | ||||||||||

390 | 390 | 390 | 398 | 398 |

**Table 5.**List of considered groups with number of conjugacy classes $cc>36$ that satisfy rule (a) (presumably (b) as well) and have been considered before as valid groups for quark/lepton mixing. A reference is given in the last column if a viable model for quark or/and lepton mixings can be obtained. The question mark means that the minimal permutation representation could not be obtained.

Group | Name or Signature | cc | Graph | Ref. |
---|---|---|---|---|

SmallGroup(726,5) | $\Delta (6\times {11}^{2})$ | 38 | ${K}_{11}^{3}$ | [15,18] |

SmallGroup(648,259) | $({\mathbb{Z}}_{18}\times {\mathbb{Z}}_{6})\u22ca{S}_{3}$, ${D}_{18,6}^{(1)}$ | 49 | ${K}_{18}^{3}$ | [2,15,18,19] |

SmallGroup(648,260) | ${\mathbb{Z}}_{3}^{2}\u22caSmallGroup(72,42)$ | 49 | ${K}_{18}^{3}$ | [2,15,18,19] |

SmallGroup(648,266) | ${\mathbb{Z}}_{3}^{2}\u22caSmallGroup(72,42)$ | 49 | ${K}_{6}^{3}$ | [15] |

SmallGroup(1176,243) | $\Delta (6\times {14}^{2})$ | 59 | ${K}_{14}^{3}$ | [15,18] |

SmallGroup(972,64) | ${\mathbb{Z}}_{9}^{2}\u22ca{\mathbb{Z}}_{12}$ | 62 | ${K}_{36}^{3}$ | [15,18] |

SmallGroup(972,245) | ${\mathbb{Z}}_{9}^{2}\u22ca({\mathbb{Z}}_{2}\times {S}_{3})$ | 62 | ${K}_{18}^{3}$ | [18] |

SmallGroup(1536,408544632) | $\Delta (6\times {16}^{2})$ | 68 | ? | [2,15,16] |

SmallGroup(1944,849) | $\Delta (6\times {18}^{2})$ | 85 | ${K}_{18}^{3}$ | [15,18] |

**Table 6.**Exceptional subgroups of $SU(3)$. For each group and each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations.

Group | d | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(60,5), $\Sigma (60)$ | 5 | 1 | 3 | 3 | 4 | 5 | |||||||

5-dit | 5 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||

(168,42), $\Sigma (168)$ | 6 | 1 | 3 | 3 | 6 | 7 | 8 | ||||||

6-dit | 6 | ${d}^{2}$ | ${d}^{2}$ | 33 | 33 | 33 | |||||||

(108,15), $\Sigma (36\times 3)$ | 14 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

14-dit | 14 | 166 | 181 | 181 | 195 | 195 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

4 | 4 | ||||||||||||

154 | 154 | ||||||||||||

(216,88), $\Sigma (72\times 3)$ | 16 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

16-dit | 16 | 175 | 175 | 157 | 233 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

2Quartits | 16 | 121 | 149 | 125 | 200 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 8 | ||||||||||

16-dit | ${d}^{2}$ | 222 | 222 | 144 | |||||||||

2Quartits | ${d}^{2}$ | 118 | 118 | 144 | |||||||||

(1080,260), $\Sigma (360\times 3)$ | 17 | 1 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 8 | 8 | 9 |

17-dit | 17 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

9 | 9 | 10 | 15 | 15 | |||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||

(648,532),$\Sigma (216\times 3)$ | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

24-dit | 24 | 527 | 527 | 562 | ${d}^{2}$ | ${d}^{2}$ | 560 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 564 | ${d}^{2}$ | ${d}^{2}$ | 552 | 552 |

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## Share and Cite

**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K.
Informationally Complete Characters for Quark and Lepton Mixings. *Symmetry* **2020**, *12*, 1000.
https://doi.org/10.3390/sym12061000

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K.
Informationally Complete Characters for Quark and Lepton Mixings. *Symmetry*. 2020; 12(6):1000.
https://doi.org/10.3390/sym12061000

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin.
2020. "Informationally Complete Characters for Quark and Lepton Mixings" *Symmetry* 12, no. 6: 1000.
https://doi.org/10.3390/sym12061000