Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C24×C32 ⋊ D4 |
| Sylow | C32 ⋊ #2846 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C22×C3⋊D3 | C24 |
| Fitting | C24×C32 | D4 |
| socle | C22×C32 | C22×D4 |
| hypercenter | C24 | GO+(4,𝔽2) |
| hypocenter | C32 | #2846 |
| LCS | G ▷ C22×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 | |
6578 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 6578 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^7,3^2|b^a=be,c^a=cf,d^a=dg,i^a=hi^2,h^d=hi,i^d=i^2,h^g=h^2,i^g=i^290 conjugacy classes — view the full character table.
| maximal subgroup of | #120×GO+(4,𝔽2), #72241536, #72241541, … |
| maximal normal subgroup of | #120×GO+(4,𝔽2), #72241536, #72241543 |
| maximal quotient of | #72253195 |
| automorphism group of | Dic32, #3722, #3724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 90180);