Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2 |
| direct product | D8×C27⋊C3 |
| unambiguous semidirect product | (D8×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C32×D4 |
| commutator | C3×C4 | C22×C3×C9 |
| Frattini, cosocle | C4×C9 | C22×C32 |
| socle | C2×C3 | C3×D4×C9 |
| prime radical | C32×D8 | C9 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C4×C27⋊C3 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8)(9,10,13,11,15,21,17,24,29,12,16,20,18,25,28,26,32,34,19,14,22,27,23,30,33,31,35);(2,8)(3,7)(4,6)(10,14,16)(13,20,22)(15,23,25)(21,28,30)(24,31,32)(29,34,35)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 607);