Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| socle | C22×C32 | C22×D4 |
| prime radical | C23×C32 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C2×C32×C42 ◁ G | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24);(4,5,6)(8,11)(12,14);(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(7,9,10,13)(8,12,11,14)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4164);