Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#14515356 |
| Sylow | C241 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11757210 |
| commutator | C2×C241 | C2×C8 |
| Fitting | C2×C4×C241 | C4 |
| socle | C22×C241 | C2×C4 |
| hypercenter | C2×C4 | #11757210 |
| hypocenter | C241 | C4⋊C8 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=d,b^2=c,d^2=e,b^a=bc,f^a=f^177,f^d=f^240500 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#14515356, #29508565, Q8⋊#14515356, … |
| maximal normal subgroup of | D4⋊#14515356, #29508565, Q8⋊#14515356, … |
| maximal quotient of | #29508578, #29508580, #29508583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 212);