Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D8×C233 ⋊ C2 |
| Sylow | C233 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D233 |
| commutator | C4×C233 | C23 |
| Frattini | C4 | C22×D233 |
| Fitting | D8×C233 | C2 |
| socle | C2×C233 | C2×D4 |
| hypercenter | D8 | D233 |
| hypocenter | C233 | #135 |
| LCS | G ▷ C4×C233 ▷ C2×C233 ▷ C233 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
62 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,233|c^2=e,d^2=e,b^a=bd,d^a=de,d^b=de,f^c=f^232826 conjugacy classes — view the full character table.
| maximal subgroup of | #28416864, #28416866, #28416869, … |
| maximal normal subgroup of | #28416864, #28416866, #28416869, … |
| maximal quotient of | #28416428, #28416472, D8×Dic233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 165);