Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1424 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | D16 |
| commutator | C16 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C16 | C22×C3 |
| socle, prime radical | C22×C3 | C3×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ C2×C9×C16 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(15,20)(16,25)(17,26)(18,29)(21,28)(22,37)(23,40)(24,34)(27,36)(30,43)(31,32)(33,38)(35,44)(39,45)(41,42);(14,15,21,35,17,22,27,33,29,38,42,37,43,44,16,20,19,24,25,32,30,39,41,40,18,23,36,45,26,31,28,34)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156);