Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1574 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C42 | C23×C32 |
| socle | C22×C32 | C2×C4⋊C4 |
| prime radical | C23×C32 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C2×C32×C42 ◁ G | |
474 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)(7,8,12,14)(9,10,13,11)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(4,5,6)(7,9,11,8,12,13,10,14)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(7,10,12,11)(8,13,14,9)(17,23)(20,26)(24,29)(27,30)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2634);