Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2145 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #102 |
| 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 | |
672 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)(9,18,21,35)(10,14)(11,25)(12,17)(13,20,29,37)(15,23)(16,24)(19,34,22,31)(26,33,38,28)(27,30)(32,36);(4,5,6)(7,9)(8,13)(10,19)(11,26)(12,21)(14,28)(15,31)(16,33)(17,29)(18,30)(20,36)(22,24)(23,37)(25,35)(27,38)(32,34);(7,10,12,24)(8,14,17,16)(9,19,21,22)(11,23,27,36)(13,28,29,33)(15,30,32,25)(18,34,35,31)(20,26,37,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3205);