Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×C22⋊C4 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| prime radical | C23×C32 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C42 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8)(9,15,17,12)(10,13)(11,14)(16,21,22,19)(18,20)(23,24,26,29,27,30,33,36)(25,28,32,35,31,34,37,38);(4,5,6)(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21)(23,25)(24,28)(26,32)(27,31)(29,35)(30,34)(33,37)(36,38);(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)(25,31)(28,34)(32,37)(35,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2530);