Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C8×C22⋊D113 |
| Fitting | C22×C8×C113 ⋊ C2 |
| Sylow | C113 ⋊ C8×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D113 |
| commutator | C2×C113 | C22×C8 |
| Frattini | C2×C4 | C22×D113 |
| Fitting | C22×C8×C113 | C2 |
| socle | C22×C113 | C22×C4 |
| hypercenter | C22×C8 | D113 |
| hypocenter | C113 | C8×D4 |
| LCS | G ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 | |
134 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1832 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #28392391, C16×C22⋊D113, #28392396, … |
| maximal normal subgroup of | #28392391, C16×C22⋊D113, #28392396, … |
| maximal quotient of | C8×#14268632, C8×#14268635, C8×#14268634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 613);