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×D4×C113 | C4 |
| hypercenter | C2×D4 | #90673 |
| hypocenter | C113 | #408 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
158 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 52 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=e,e^2=f,b^a=bd,g^a=g^98,c^b=cf,g^e=g^112308 conjugacy classes — view the full character table.
| maximal subgroup of | #28405661, #28405664, #28406419, … |
| maximal normal subgroup of | #28405661, #28405664, #28406419, … |
| maximal quotient of | #28405839, #28405841, #28405843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1522);