Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×#1366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #333 |
| commutator | C42 | C2×C32×C4 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| prime radical | C22×C32 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×C4×C8 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3)(7,8,10,15,23,16,34,32,12,17,25,33,11,14,24,30)(9,18,20,31,26,29,22,37,19,35,36,38,21,13,27,28);(4,5,6)(7,9,12,19)(8,13)(10,22,25,27)(11,26,23,21)(14,18)(15,31)(16,35)(17,29)(20,24,36,34)(28,32)(30,37)(33,38)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2426);