Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C113 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C22⋊Q8×C113 | C2 |
| hypercenter | C22⋊Q8 | D113 |
| hypocenter | C113 | #541 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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,b^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce,g^b=g^112806 conjugacy classes — view the full character table.
| maximal subgroup of | #28400781, #28400783, #28400789, … |
| maximal normal subgroup of | #28400781, #28400783, #28400789, … |
| maximal quotient of | #28393181, #28393219, #28393221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1159);