Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C113 ⋊ #408 |
| 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 | #408 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
158 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,113|a^2=e,e^2=f,b^a=bd,g^a=g^98,c^b=cf,g^b=g^112,g^c=g^112,g^e=g^112308 conjugacy classes — view the full character table.
| maximal subgroup of | #28405676, #28406423, #28406532, … |
| maximal normal subgroup of | #28405676, #28406423, #28406532, … |
| maximal quotient of | #28405842, #28405845, #28405848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1524);