Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#14272347 |
| Fitting | C4×C233 ⋊ C8 |
| Sylow | C233 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#14272347 |
| commutator, socle | C2×C233 | C2×C8 |
| Fitting | C4×C233 | C8 |
| hypercenter | C4 | #14272347 |
| hypocenter | C233 | C4⋊C8 |
| LCS | G ▷ C2×C233 ▷ C233 | |
| UCS | C1 ◁ 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^221,f^d=f^144,f^e=f^232136 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#14272347, Q8⋊#14272347, #28417524, … |
| maximal normal subgroup of | D4⋊#14272347, Q8⋊#14272347, #28417524, … |
| maximal quotient of | #28417524, #28417525, (C2×#14272347)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 239);