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) = C3×C4 |
| direct product | C32×#1922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C22×C32 | C4×D4 |
| prime radical | C23×C32 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C22×C32×C8 ◁ G | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,19,23,20,18,27,22)(17,21,25,29,26,24,30,28);(4,5,6)(8,11)(12,14)(15,17,19,25,20,26,27,30)(16,21,23,29,18,24,22,28);(7,9,10,13)(8,12,11,14)(15,18)(16,20)(17,24)(19,22)(21,26)(23,27)(25,28)(29,30)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2982);