Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | (C2×#90673)⋊Q8 |
| Fitting | C2×Q8×C113 ⋊ C4 |
| Sylow | C113 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#90673 |
| commutator, socle | C22×C113 | C22×C4 |
| Fitting | C2×Q8×C113 | C4 |
| hypercenter | C2×Q8 | #90673 |
| hypocenter | C113 | #391 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,113|a^2=f,b^2=def,c^2=e,b^a=bd,c^a=ce,g^a=g^15,g^b=g^98,g^f=g^112308 conjugacy classes — view the full character table.
| maximal subgroup of | #28405670, #28406421, #28406530, … |
| maximal normal subgroup of | #28405670, #28406421, #28406530, … |
| maximal quotient of | #28405751, #28405753, #28405759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1502);