Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C42 |
| commutator | C4 | C4×C8×C9 |
| Frattini, cosocle | C2×C3×C16 | C22×C3 |
| socle | C2×C3 | C3×#346 |
| prime radical | C22×C3 | C3×M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,17,13,19,22,32,14,20,23,30,24,34,36,40,15,18,21,31,25,33,35,41,26,16,27,28,37,29,38,39);(11,16,18,20)(12,21)(17,30,31,28)(19,29,33,34)(22,35)(23,27)(32,40,41,39)(36,38)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 130);