Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C113 ⋊ C2 |
| Sylow | C113 ⋊ #552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C22⋊Q8×C113 | C2 |
| hypercenter | C22⋊Q8 | D113 |
| hypocenter | C113 | #552 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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,c^2=e,b^a=be,c^a=cf,g^a=g^112,d^b=de,g^b=g^112,g^c=g^112809 conjugacy classes — view the full character table.
| maximal subgroup of | #28402413, #28402415, #28402465, … |
| maximal normal subgroup of | #28402413, #28402415, #28402465, … |
| maximal quotient of | #28393432, #28393445, #28393458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1241);