Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#3131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C3×#532 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C2×C42×C9 ◁ G | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19)(20,26)(21,25)(22,27)(23,28)(24,29)(30,33)(31,32);(11,14)(15,17)(18,20,23,30,24,31,22,21)(19,25,27,32,29,33,28,26);(10,12,13,16)(11,15,14,17)(18,21,22,31,24,30,23,20)(19,26,28,33,29,32,27,25);(18,22,24,23)(19,27,29,28)(20,21,31,30)(25,32,33,26)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1863);