Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3298 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C4×D4 ◁ G | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,10,12)(9,13,14,11)(16,19)(17,21)(20,22);(4,5,6)(7,9)(8,11)(10,14)(12,13);(15,16,18,20)(17,22,21,19);(9,14)(11,13)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4358);