Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C32×#1309 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #95 |
| commutator | C23 | C32×C42 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| prime radical | C23×C32 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C42 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,9,12,10,11,13,14)(16,19)(17,21);(4,5,6)(8,11)(9,13)(15,16,17,20,18,19,21,22)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2369);