Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×#1366 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #333 |
| commutator | C42 | C2×C4×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle | C2×C3 | C3×#333 |
| prime radical | C22×C3 | C3×#100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C4×C8×C9 ◁ G | |
111 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)(10,11,13,18,26,19,37,35,15,20,28,36,14,17,27,33)(12,21,23,34,29,32,25,40,22,38,39,41,24,16,30,31);(10,12,15,22)(11,16)(13,25,28,30)(14,29,26,24)(17,21)(18,34)(19,38)(20,32)(23,27,39,37)(31,35)(33,40)(36,41)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 98);