Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#3223 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,16)(13,15)(14,17)(18,19,23,27)(20,24,29,38)(21,31,33,41)(22,34,36,26)(25,35,40,32)(28,42,43,49)(30,46,45,39)(37,47,48,44);(10,12,13,17)(11,14,15,16)(18,20)(19,24)(21,28)(22,30)(23,29)(25,37)(26,39)(27,38)(31,42)(32,44)(33,43)(34,46)(35,47)(36,45)(40,48)(41,49);(12,17)(14,16)(18,21,23,33)(19,25,27,40)(20,28,29,43)(22,35,36,32)(24,37,38,48)(26,41,34,31)(30,47,45,44)(39,49,46,42);(11,15)(14,16)(20,29)(24,38)(28,43)(30,45)(37,48)(39,46)(42,49)(44,47)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1955);