Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C32×#41 ⋊ D4 |
| Sylow | C32 ⋊ #3498 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C25 |
| Fitting | C32×#41 | D4 |
| socle | C2×C32 | C23×D4 |
| hypercenter | #41 | GO+(4,𝔽2) |
| hypocenter | C32 | #3498 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
2646 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 2646 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^7,3^2|c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,i^a=hi^2,d^b=df,h^c=hi,i^c=hi^2,h^g=h^2,i^g=i^278 conjugacy classes — view the full character table.
| maximal subgroup of | #72220444, H(5,𝔽2)×GO+(4,𝔽2), #72222913 |
| maximal normal subgroup of | #72220444, H(5,𝔽2)×GO+(4,𝔽2), #72222913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 140006);