Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4⋊Q8×C113 ⋊ C2 |
| Sylow | C113 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C4⋊Q8×C113 | C2 |
| hypercenter | C4⋊Q8 | D113 |
| hypocenter | C113 | #533 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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=f,c^2=f,d^2=e,b^a=be,c^a=cf,g^b=g^112,g^d=g^112812 conjugacy classes — view the full character table.
| maximal subgroup of | #28389228, #28389230, #28389237, … |
| maximal normal subgroup of | #28389228, #28389230, #28389237, … |
| maximal quotient of | #28392979, #28393045, #28393049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1104);