Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2148 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #102 |
| commutator | C8 | C22×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C23×C3 |
| socle | C2×C3 | C2×C3×#102 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ C22×C8×C9 ◁ G | |
330 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)(10,11,13,17,15,20,27,19)(12,21,24,35,25,38,22,37)(14,28,26,18,30,34,39,36)(16,29,32,23,33,40,31,41);(10,12)(11,16)(13,22)(14,29)(15,25)(17,31)(18,35)(19,32)(20,33)(21,34)(23,39)(24,27)(26,41)(28,38)(30,40)(36,37);(10,13,15,27)(11,17,20,19)(12,22,25,24)(14,26,30,39)(16,31,33,32)(18,34,36,28)(21,37,38,35)(23,29,41,40)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 880);