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×#2148 |
| 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 | |
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)(7,8,10,14,12,17,24,16)(9,18,21,32,22,35,19,34)(11,25,23,15,27,31,36,33)(13,26,29,20,30,37,28,38);(4,5,6)(7,9)(8,13)(10,19)(11,26)(12,22)(14,28)(15,32)(16,29)(17,30)(18,31)(20,36)(21,24)(23,38)(25,35)(27,37)(33,34);(7,10,12,24)(8,14,17,16)(9,19,22,21)(11,23,27,36)(13,28,30,29)(15,31,33,25)(18,34,35,32)(20,26,38,37)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3208);