Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C241 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16838755 |
| commutator | C8×C241 | C22 |
| Frattini | C8 | C2×D241 |
| Fitting | Q16×C241 | C2 |
| socle | C2×C241 | D8 |
| hypercenter | Q16 | D241 |
| hypocenter | C241 | Q32 |
| LCS | G ▷ C8×C241 ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,c^b=cd,d^b=de,f^b=f^240851 conjugacy classes — view the full character table.
| maximal subgroup of | Q32×D241, C2×#21440588, #29508147, … |
| maximal normal subgroup of | Q32×D241, C2×#21440588, #29508147, … |
| maximal quotient of | #29507520, #29507521, #29507542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 92);