Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#2253 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 46 points:
(1,2,3)(7,8,10,12)(9,13,14,11)(15,16,18,21)(17,25,27,39)(19,32,30,45)(20,33,31,24)(22,36,34,28)(23,38,35,26)(29,44,43,41)(37,42,40,46);(4,5,6)(7,9,10,14)(8,11,12,13)(16,21)(22,34)(23,35)(24,33)(25,39)(32,45)(37,40)(41,44);(15,17,26,42,20,29,30,36,18,27,38,46,31,43,19,28)(16,22,32,41,24,37,35,39,21,34,45,44,33,40,23,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3313);