Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C32 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C25 |
| Fitting | C2×C32×Q8 | D4 |
| socle | C2×C32 | C23×D4 |
| hypercenter | C2×Q8 | GO+(4,𝔽2) |
| hypocenter | C32 | #3584 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
2234 conjugacy classes of subgroups; only the 443 normal subgroups are shown.
This group has 2234 conjugacy classes of subgroups — too many to draw. Only its 443 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^7,3^2|e^2=g,f^2=g,b^a=bf,f^a=fg,i^a=hi^2,f^b=fg,h^b=hi,i^b=i^2,d^c=dg,h^c=h^2,i^c=i^2,h^d=h^2,i^d=i^2,h^e=h^2,i^e=i^2,h^f=h^2,i^f=i^278 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153027);