Order = 5515776 = 2^{9}.3^{4}.7.19.

Mult = 3.

Out = 3 × S_{3}.

## Porting notes

Porting incomplete.## Standard generators

Standard generators of U_{3}(8) are *a*, *b* where *a* has order 2, *b* has order 3 (necessarily in class 3C) and *a**b* has order 19.

Standard generators of 3.U_{3}(8) = SU_{3}(8) are preimages *A*, *B* where *A* has order 2 and *A**B* has order 19.

Standard generators of U_{3}(8):2 are *c*, *d* where *c* is in class 2B, *d* is in class 3C, *c**d* has order 8 and *c**d**c**d**c**d**d**c**d**c**d**d**c**d**d* has order 9.

Standard generators of 3.U_{3}(8):2 are preimages *C*, *D* where *C**D**C**D**D* has order 19.

Standard generators of U_{3}(8):3_{1} are *e*, *f* where *e* has order 2, *f* is in class 3D/E/F/D'/E'/F', *e**f* has order 12, *e**f**e**f**f* has order 7 and *e**f**e**f**e**f**f**e**f**e**f**f**e**f**f* has order 7.

Standard generators of 3.U_{3}(8):3_{1} are preimages *E*, *F* where *E* has order 2 and *F* has order 3.

Standard generators of U_{3}(8):6 are *g*, *h* where *g* is in class 2B, *h* is in class 3D/D'/EF/EF' (i.e. an outer element of order 3), *g**h* has order 18, *g**h**g**h**h* has order 19 and *g**h**g**h**g**h**h**g**h**g**h**h**g**h**h* has order 9.

Standard generators of U_{3}(8):3_{2} are *i*, *j* where *i* has order 2, *j* is in class 3G/G' and *i**j* has order 9.

Standard generators of U_{3}(8).3_{3} are *k*, *l* where *k* has order 2, *l* is in class 9K/L/M/K'/L'/M', *k**l* has order 9, *k**l*^{2} has order 9, *k**l*^{3} has order 6, *k**l*^{4} has order 18 and *k**l**k**l*^{2}*k**l*^{4} has order 9.

Standard generators of U_{3}(8):S_{3} are *m*, *n* where *m* is in class 2B, *n* is in class 3G/G', *m**n* has order 8, *m**n**m**n**n* has order 9 and (*m**n*)^{3}*m**n*^{2}*m**n**m**n*^{2}*m**n*^{2} has order 14.

Standard generators of U_{3}(8).3^{2} are *o*, *p* where *o* is in class 3DEF/DEF', *p* is in class 9EFG/EFG', *o**p* has order 9, *o**p*^{2} has order 9, *o**p*^{3} has order 12, *o**p*^{4} has order 9 and *o**p**o*^{2}*p*^{2} has order 7.

Standard generators of U_{3}(8).(S_{3} × 3) are *q*, *r* where *q* is in class 2B, *r* is in class 9KLM/KLM', *q**r* has order 6, *q**r**q**r**r* has order 3 and *q**r**q**r*^{2}*q**r*^{4} has order 6.

## Presentations

Group | Presentation | Link |
---|---|---|

U_{3}(8)
| 〈 a, b | a^{2} = b^{3} = (ab)^{19} = [a,b]^{9} = [a,bab]^{3} = (abababab^{−1})^{3}ab^{−1}ab(ab^{−1})^{3}ab(ab^{−1})^{2} = (((ab)^{4}(ab^{−1})^{3})^{2}ab^{−1})^{2} = 1 〉
| Details |

## Representations

### Representations of U_{3}(8)

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Q(ω) 57 a Std Details 0 Q(ω) 57 b Std Details 0 Z 114 Std Details 0 Z 133 a Std Details 0 Z 133 b Std Details 0 Z 133 c Std Details 3 GF(3) 56 Std Details

### Representations of 3.U_{3}(8)

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- Permutation representations:
Number of points ID Generators Description Link 4617 Std Details 32832 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(64) 3 a Std Details

### Representations of U_{3}(8):2

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8):3_{1}

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8):6

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 24 Std Details 2 GF(2) 54 a Std Details 2 GF(2) 54 b Std Details 2 GF(2) 192 Std Details 2 GF(2) 432 Std Details 2 GF(2) 512 Std Details 3 GF(3) 56 a Std Details 3 GF(3) 133 a Std Details 3 GF(3) 266 Std Details

### Representations of U_{3}(8):3_{2}

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8).3_{3}

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8):S_{3}

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8).3^{2}

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

### Representations of U_{3}(8).(S_{3} × 3)

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- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details