Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #120 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #385 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8,12,16,13,17,11,24)(9,18,20,14,21,29,25,28)(10,15,22,31,23,32,26,30)(19,34,37,38,36,33,35,27);(4,5,6)(7,9,12,20,13,21,11,25)(8,14,16,29,17,28,24,18)(10,19,22,37,23,36,26,35)(15,27,31,34,32,38,30,33);(7,10)(8,15)(9,19)(11,26)(12,22)(13,23)(14,27)(16,31)(17,32)(18,33)(20,37)(21,36)(24,30)(25,35)(28,38)(29,34)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2918);