Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C22⋊#11759967 |
| unambiguous semidirect product | D257⋊(C22⋊C4) |
| unambiguous semidirect product | (C2×#11759967)⋊C22 |
| Fitting | C23×C257 ⋊ C4 |
| Sylow | C257 ⋊ C2×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#11759967 |
| commutator | C2×C257 | C22×C4 |
| Frattini | C2 | C22×#11759967 |
| Fitting | C23×C257 | C4 |
| socle | C22×C257 | C2×C4 |
| hypercenter | C23 | #11759967 |
| hypocenter | C257 | C2×C22⋊C4 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ C23 | |
132 conjugacy classes of subgroups.
532 conjugacy classes — view the full character table.
| maximal subgroup of | #30657614, #30657615, C4×C22⋊#11759967, … |
| maximal normal subgroup of | #30657614, #30657615, C4×C22⋊#11759967, … |
| maximal quotient of | C2×(C2×#11759967)⋊C4, C4×C22⋊#11759967, #30657678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 223);