Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C241 ⋊ #108 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#11757210 |
| commutator | C4×C241 | C2×C4 |
| Frattini | C4 | C2×#11757210 |
| Fitting | C8×C241 | C4 |
| socle | C2×C241 | C4⋊C4 |
| hypercenter | C8 | #11757210 |
| hypocenter | C241 | #108 |
| LCS | G ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
30 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bc,c^a=ce,f^a=f^177,f^b=f^240,f^d=f^240494 conjugacy classes — view the full character table.
| maximal subgroup of | #29508631, #29508632, #29508645, … |
| maximal normal subgroup of | #29508631, #29508632, #29508645, … |
| maximal quotient of | #29508579, #29508580, #29508593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 217);