Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#2247 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2,3)(8,10)(11,12,16,20)(13,21,25,37)(14,26,28,41)(15,29,27,19)(17,30,32,23)(18,33,34,22)(24,40,39,36)(31,38,35,42);(4,5,6)(11,13,22,38,15,24,28,30,16,25,33,42,27,39,14,23)(12,17,26,36,19,31,34,37,20,32,41,40,29,35,18,21);(7,8,9,10)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3307);