Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #3055 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊Dic3 | C24 |
| Fitting | C24×C32 | D4 |
| socle | C22×C32 | C22×D4 |
| hypercenter | C24 | GO+(4,𝔽2) |
| hypocenter | C32 | #3055 |
| LCS | G ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 | |
926 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 926 conjugacy classes of subgroups — too many to draw. Only its 115 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|b^2=e,e^2=g,b^a=be,c^a=cf,e^a=eg,i^a=hi^2,d^b=dg,h^b=hi,i^b=hi^2,h^e=h^2,i^e=i^266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 101213);