Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C113 ⋊ C4 |
| Sylow | C113 ⋊ #444 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#90673 |
| commutator | C4×C113 | C22×C4 |
| Frattini | C4 | C22×#90673 |
| Fitting | Q16×C113 | C4 |
| socle | C2×C113 | C4×D4 |
| hypercenter | Q16 | #90673 |
| hypocenter | C113 | #444 |
| LCS | G ▷ C4×C113 ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
104 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,113|a^2=e,c^2=e,d^2=f,e^2=f,b^a=bd,c^a=cf,d^a=df,g^a=g^98,d^b=df,g^b=g^112,g^c=g^98,g^e=g^112218 conjugacy classes — view the full character table.
| maximal subgroup of | #28406981, #28406982, #28406985, … |
| maximal normal subgroup of | #28406981, #28406982, #28406985, … |
| maximal quotient of | #28406164, #28406167, #28406172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1595);