Order = 3732063900024176640 = 2^{35}.3^{8}.5.7.11.43.

Mult = 1.

Out = 2.

## Porting notes

Porting incomplete.## Standard generators

Standard generators of 2^{14}.U_{7}(2) are *a*, *b* where *a* is in class 2C, *b* has order 7 and *a**b* has order 33.

### Notes

- (2
^{14}.U_{7}(2)) 2^{14}.U_{7}(2) has a unique conjugacy class of subgroups of index 10836, and we may take*a*to be a central involution in such a subgroup. In fact, this subgroup turns out to be C(*a*). Also, a happens to be in a class of 3-transpositions, and a commutes with 2644 of its conjugates. The orbits of C(*a*) (acting by conjugation) on the conjugates of a have sizes 1, 3, 2640 and 8192. - (2
^{14}.U_{7}(2)) No extra conditions (such as*abb*having order 45) are required. These generators map onto standard generators of U_{7}(2).

## Presentations

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

2^{14}.U_{7}(2)
| 〈 a, b | a^{2} = b^{7} = (ab)^{33} = [a,b]^{3} = [a,b^{2}]^{3} = [a,b^{3}]^{3} = [a,bab]^{2} = [a,b^{2}ab^{2}]^{2} = [a,bab^{2}]^{3} = (abab^{3}abab^{−2}ab^{−2})^{6} = 1 〉
| Details |

## Representations

### Representations of 2^{14}.U_{7}(2)

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 10836 Std Details

## Miscellaneous Notes

Group | Category | Note |
---|---|---|

2^{14}.U_{7}(2)
| Bibliographic reference. | As far as we know, this group was discovered by J.I.Hall. He also discovered its 3-transposition property. |