Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1521 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C2×C3×C22⋊C4 |
| prime radical | C3×#121 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C42×C9 ◁ G | |
327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(11,16,18,30)(12,21)(13,26)(17,32,33,38)(19,31,20,27)(22,37,34,29)(24,40)(25,36)(28,39)(35,41);(10,11)(12,22)(13,27)(14,19)(15,20)(16,28)(17,25)(18,23)(21,33)(24,37)(26,30)(29,40)(31,39)(32,41)(34,36)(35,38);(10,12,23,36)(11,17,18,33)(13,24,39,40)(14,25,15,21)(16,29,30,37)(19,22,20,34)(26,41,28,35)(27,32,31,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 253);