Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×#117 ◁ G | |
1038 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)(8,11)(9,13)(15,16,18,22)(17,24,25,20)(19,21,27,23)(26,28,30,29);(4,5,6)(7,8)(9,12)(10,11)(13,14)(15,17,18,25)(16,20,22,24)(19,26,27,30)(21,29,23,28);(8,11)(12,14)(16,21)(17,25)(19,27)(20,28)(22,23)(24,29);(7,9,10,13)(8,12,11,14)(16,22)(17,25)(21,23)(26,30)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4323);