Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22⋊Q8×C113 ⋊ C2 |
| Sylow | C113 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C22⋊Q8×C113 | C2 |
| hypercenter | C22⋊Q8 | D113 |
| hypocenter | C113 | #536 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,113|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce,g^b=g^112,g^d=g^112806 conjugacy classes — view the full character table.
| maximal subgroup of | #28399846, #28399848, #28399864, … |
| maximal normal subgroup of | #28399846, #28399848, #28399864, … |
| maximal quotient of | #28393157, #28393163, #28393196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1122);