Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C22⋊#11757210 |
| unambiguous semidirect product | D241⋊(C22⋊C4) |
| unambiguous semidirect product | (C2×#11757210)⋊C22 |
| Fitting | C23×C241 ⋊ C4 |
| Sylow | C241 ⋊ C2×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#11757210 |
| commutator | C2×C241 | C22×C4 |
| Frattini | C2 | C22×#11757210 |
| Fitting | C23×C241 | C4 |
| socle | C22×C241 | C2×C4 |
| hypercenter | C23 | #11757210 |
| hypocenter | C241 | C2×C22⋊C4 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C23 | |
132 conjugacy classes of subgroups.
500 conjugacy classes — view the full character table.
| maximal subgroup of | #29508594, #29508595, C4×C22⋊#11757210, … |
| maximal normal subgroup of | #29508594, #29508595, C4×C22⋊#11757210, … |
| maximal quotient of | C2×(C2×#11757210)⋊C4, C4×C22⋊#11757210, #29508658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 223);