Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | Q8×#11757210 |
| Fitting | Q8×C241 ⋊ C4 |
| Sylow | C241 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#11757210 |
| commutator, socle | C2×C241 | C22×C4 |
| Fitting | Q8×C241 | C4 |
| hypercenter | Q8 | #11757210 |
| hypocenter | C241 | C4×Q8 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
70 conjugacy classes of subgroups.
320 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×#11757210, Q16×#11757210, #29508756, … |
| maximal normal subgroup of | SD16×#11757210, Q16×#11757210, #29508756, … |
| maximal quotient of | C4⋊C4×#11757210, #29508666, #29508668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 227);