Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ C4×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#385 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #385 | D53 |
| hypocenter | C53 | C4×C22⋊Q8 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #385 | |
656 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=e,d^2=eg,b^a=be,c^a=cf,h^b=h^52,h^d=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C22⋊Q8×D53, C2×#20277247, C4×#16811129, … |
| maximal normal subgroup of | C4×C22⋊Q8×D53, C2×#20277247, C4×#16811129, … |
| maximal quotient of | #27305256, #385×Dic53, #27305266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7042);