Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#1411 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #353 |
| commutator | C42 | C2×C4×C9 |
| Frattini, cosocle | C3×C4⋊Q8 | C22×C3 |
| socle | C2×C3 | C3×#353 |
| prime radical | C22×C3 | C3×#99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×C4⋊Q8 ◁ G | |
141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,18,15,20,27,37)(12,21,22,35,24,32,39,40)(14,19,29,36,30,17,26,34)(16,25,31,28,33,38,41,23);(10,12,15,24)(11,16,20,33)(13,25,27,38)(14,28,30,23)(17,32,19,21)(18,35,37,40)(22,26,39,29)(31,36,41,34)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 143);