Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#14515358 |
| Fitting | C4×C241 ⋊ C8 |
| Sylow | C241 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#14515358 |
| commutator, socle | C2×C241 | C2×C8 |
| Fitting | C4×C241 | C8 |
| hypercenter | C4 | #14515358 |
| hypocenter | C241 | C4⋊C8 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ 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^30,f^d=f^177,f^e=f^240140 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#14515358, Q8⋊#14515358, #29508806, … |
| maximal normal subgroup of | D4⋊#14515358, Q8⋊#14515358, #29508806, … |
| maximal quotient of | #29508806, #29508807, (C2×#14515358)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 239);