Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C113 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D113 |
| commutator | C22×C113 | C22×C4 |
| Fitting | #95×C113 | C2 |
| socle | C23×C113 | C23 |
| hypercenter | #95 | D113 |
| hypocenter | C113 | #387 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C23 ◁ #95 | |
154 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 69 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=f,c^2=d,b^a=bd,c^a=ce,g^b=g^112,g^c=g^1121148 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #387×D113, C2×#20298481, #28393014, … |
| maximal normal subgroup of | #387×D113, C2×#20298481, #28393014, … |
| maximal quotient of | #28388800, #28388805, #28388808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 432);