Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | D8 |
| commutator | C8 | C42×C9 |
| Frattini, cosocle | C22×C3×C8 | C22×C3 |
| socle | C22×C3 | C3×#102 |
| prime radical | C23×C3 | C3×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C4×C9 ◁ C22×C8×C9 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(14,15)(16,25)(17,30)(18,33)(19,35)(20,24)(21,26)(22,32)(23,31)(27,37)(28,45)(29,41)(34,38)(36,44)(39,43)(40,42);(14,16,18,27,31,42,32,43,20,29,17,26,19,28,34,44)(15,21,22,25,35,39,33,45,24,37,38,41,23,36,30,40)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 114);