Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#14272345 |
| Sylow | C233 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11745400 |
| commutator | C2×C233 | C2×C8 |
| Fitting | C2×C4×C233 | C4 |
| socle | C22×C233 | C2×C4 |
| hypercenter | C2×C4 | #11745400 |
| hypocenter | C233 | C4⋊C8 |
| LCS | G ▷ C2×C233 ▷ C233 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,233|a^2=d,b^2=c,d^2=e,b^a=bc,f^a=f^144,f^d=f^232484 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#14272345, #28417283, Q8⋊#14272345, … |
| maximal normal subgroup of | D4⋊#14272345, #28417283, Q8⋊#14272345, … |
| maximal quotient of | #28417296, #28417298, #28417301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 212);