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)

- View detailed report.
- 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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- 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 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}

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details

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

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details