Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C233 ⋊ C2 |
| Sylow | C233 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D233 |
| commutator | C4×C233 | C23 |
| Frattini | C4 | C22×D233 |
| Fitting | Q16×C233 | C2 |
| socle | C2×C233 | C2×D4 |
| hypercenter | Q16 | D233 |
| hypocenter | C233 | #137 |
| LCS | G ▷ C4×C233 ▷ C2×C233 ▷ C233 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,233|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^232,d^b=de,f^c=f^232823 conjugacy classes — view the full character table.
| maximal subgroup of | #137×D233, C2×#21418567, #28417232, … |
| maximal normal subgroup of | #137×D233, C2×#21418567, #28417232, … |
| maximal quotient of | #28416435, #28416442, #28416476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 178);