Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×C22⋊D113 |
| Fitting | C22×Q8×C113 ⋊ C2 |
| Sylow | C113 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C22×Q8×C113 | C2 |
| hypercenter | C22×Q8 | D113 |
| hypocenter | C113 | D4×Q8 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1145 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #28402077, #28402078, #28402084, … |
| maximal normal subgroup of | #28402077, #28402078, #28402084, … |
| maximal quotient of | Q8×#14268624, Q8×C22⋊Dic113, C4⋊C4×C22⋊D113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1213);