Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×C7×C27⋊C3 |
| unambiguous semidirect product | (C2×C27)⋊(C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C9 | C32 |
| commutator | C3 | C2×C3×C7×C9 |
| Frattini, cosocle | C9 | C2×C32×C7 |
| socle | C2×C3×C7 | C3×C9 |
| prime radical | C2×C32×C7 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C9 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 378 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4,5,6,7,8,9)(10,11,14,12,16,22,18,25,30,13,17,21,19,26,29,27,33,35,20,15,23,28,24,31,34,32,36);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36)462 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 132);