Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D241⋊M16 |
| Sylow | C241 ⋊ C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#11757210 |
| commutator, socle | C2×C241 | C22×C4 |
| Frattini | C2 | C22×#11757210 |
| Fitting | C2×C4×C241 | C4 |
| hypercenter | C2×C4 | #11757210 |
| hypocenter | C241 | C2×M16 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
68 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=d,d^2=e,b^a=be,f^a=f^177,f^c=f^240,f^d=f^240500 conjugacy classes — view the full character table.
| maximal subgroup of | #29508584, #29508586, #29508588, … |
| maximal normal subgroup of | #29508584, #29508586, #29508588, … |
| maximal quotient of | #29508676, C4×#16838779, C4×#16838780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 231);