Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C241 ⋊ C2 |
| Sylow | C241 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D241 |
| commutator | C4×C241 | C23 |
| Frattini | C4 | C22×D241 |
| Fitting | Q16×C241 | C2 |
| socle | C2×C241 | C2×D4 |
| hypercenter | Q16 | D241 |
| hypocenter | C241 | #137 |
| LCS | G ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^240,d^b=de,f^c=f^240851 conjugacy classes — view the full character table.
| maximal subgroup of | #137×D241, C2×#21440674, #29508514, … |
| maximal normal subgroup of | #137×D241, C2×#21440674, #29508514, … |
| maximal quotient of | #29507717, #29507724, #29507758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 178);