Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| prime radical | C2×C32×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8)(9,11)(10,13)(12,14)(15,16,21,25)(17,26,18,29)(19,24,20,23)(22,28,30,27);(4,5,6)(8,11)(9,14)(12,13)(15,17,21,18)(16,22,25,30)(19,28,20,27)(23,29,24,26);(15,18,21,17)(16,22,25,30)(19,27,20,28)(23,29,24,26)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2709);