Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C233 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16836579 |
| commutator | C8×C233 | C22 |
| Frattini | C8 | C2×D233 |
| Fitting | Q16×C233 | C2 |
| socle | C2×C233 | D8 |
| hypercenter | Q16 | D233 |
| hypocenter | C233 | Q32 |
| LCS | G ▷ C8×C233 ▷ C4×C233 ▷ C2×C233 ▷ C233 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,233|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,c^b=cd,d^b=de,f^b=f^232823 conjugacy classes — view the full character table.
| maximal subgroup of | Q32×D233, C2×#21418481, #28416865, … |
| maximal normal subgroup of | Q32×D233, C2×#21418481, #28416865, … |
| maximal quotient of | #28416238, #28416239, #28416260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 92);