Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C2×D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | C2×#102 |
| prime radical | C2×C32×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
660 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)(7,8,11,16,13,17,12,27)(9,18,21,35,19,34,28,32)(10,22,24,33,26,37,25,15)(14,23,30,36,29,38,31,20);(4,5,6)(7,9)(8,14)(10,23)(11,21)(12,28)(13,19)(15,32)(16,30)(17,29)(18,22)(20,25)(24,36)(26,38)(27,31)(33,35)(34,37);(9,19)(14,29)(18,34)(20,36)(21,28)(23,38)(30,31)(32,35)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3203);