Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C32×D4 ⋊ D4 |
| Sylow | C32 ⋊ #2668 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C22×C3⋊D3 | C24 |
| Fitting | C2×C32×D4 | D4 |
| socle | C22×C32 | C22×D4 |
| hypercenter | C2×D4 | GO+(4,𝔽2) |
| hypocenter | C32 | #2668 |
| LCS | G ▷ C22×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
1360 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^7,3^2|b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,i^b=hi^2,h^c=hi,i^c=i^2,h^g=h^2,i^g=i^260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 70020);