Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#2183 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C8 | C23 |
| commutator | C22 | C22×C32×C8 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle, prime radical | C22×C32 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C8 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8)(9,14)(10,12)(11,13)(15,16,18,22,19,17,24,25,20,23,27,30,28,26,21,29);(4,5,6)(7,9,11,10)(8,12,13,14);(7,10,11,9)(8,12,13,14)(15,17,18,25,19,23,24,30,20,26,27,29,28,16,21,22)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3243);