Finite Group DatabasebetaFinite Group Databasebeta| order | 7856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C23×D491 |
| unambiguous semidirect product | C491⋊C24 |
| unambiguous semidirect product | (C2×C491)⋊C23 |
| Fitting, socle | C23×C491 ⋊ C2 |
| derived, Sylow | C491 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D491 |
| commutator, hypocenter | C491 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C491 | C2 |
| LCS | G ▷ C491 | |
| UCS | C1 ◁ C23 | |
134 conjugacy classes of subgroups.
1976 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22⋊C4×D491, C2×(C2×D491)⋊C4, #30628305, … |
| maximal normal subgroup of | C22⋊C4×D491, C2×(C2×D491)⋊C4, #30628305, … |
| maximal quotient of | C22×C4×D491, C23×Dic491, C2×D4×D491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7856, 42);