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 |
| direct product | C32×#2327 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C24 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(8,10)(11,12,14,13)(16,20)(17,23)(18,26)(28,30);(4,5,6)(11,13,14,12)(15,16)(17,24)(18,22)(19,20)(21,23)(25,30)(26,27)(28,29);(7,8,9,10)(15,17)(16,21)(18,25)(19,23)(20,24)(22,28)(26,29)(27,30);(15,18,19,26)(16,22,20,27)(17,25,23,29)(21,28,24,30)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3387);