Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #121 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#390 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,16,21,30)(13,23)(14,26)(15,18)(17,24)(19,27)(20,32,33,41)(22,36,35,29)(25,37)(28,39,40,34)(31,38);(10,12)(11,16,18,30)(13,20)(14,22)(15,21)(17,28,31,40)(19,29,26,36)(23,32,37,41)(24,34)(25,33)(27,35)(38,39);(10,13,14,24,15,25,27,38)(11,17,19,23,18,31,26,37)(12,20,22,34,21,33,35,39)(16,28,29,32,30,40,36,41)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 708);