Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | Q16×#90673 |
| Fitting | Q16×C113 ⋊ C4 |
| Sylow | C113 ⋊ C4×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#90673 |
| commutator | C4×C113 | C22×C4 |
| Frattini | C4 | C22×#90673 |
| Fitting | Q16×C113 | C4 |
| socle | C2×C113 | C4×D4 |
| hypercenter | Q16 | #90673 |
| hypocenter | C113 | C4×Q16 |
| LCS | G ▷ C4×C113 ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
110 conjugacy classes of subgroups.
224 conjugacy classes — view the full character table.
| maximal subgroup of | SD32×#90673, Q32×#90673, #28406970, … |
| maximal normal subgroup of | SD32×#90673, Q32×#90673, #28406970, … |
| maximal quotient of | Q16×#11744753, #28406161, #28406162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1587);