Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C25×C3×C4 |
| direct product | C32×#2310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| prime radical | C24×C32 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
2352 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)(8,10)(11,12,13,14)(16,20)(17,23)(18,26)(28,30);(4,5,6)(15,16,19,20)(17,24,23,21)(18,22,26,27)(25,30,29,28);(7,8)(9,10)(11,12)(13,14)(15,17,19,23)(16,21,20,24)(18,25,26,29)(22,28,27,30);(15,18)(16,22)(17,25)(19,26)(20,27)(21,28)(23,29)(24,30)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3370);