Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1296 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22⋊C4 |
| commutator | C2×C4 | C42×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle, prime radical | C23×C3 | C3×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C2×C42×C9 ◁ G | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(15,19)(16,17)(20,21)(22,23)(24,26)(25,28)(27,29);(14,15,17,20,18,21,16,19)(23,26)(24,29)(27,28)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 28);