Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C23×C32 |
| direct product | C2×C3×D4×3−1+2 |
| unambiguous semidirect product | C4⋊(C22×C3×3−1+2) |
| unambiguous semidirect product | (C2×D4×C9)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C23×C33 |
| prime radical | C2×C33×D4 | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(7,8,9);(3,4,5,6)(10,11,13,12,15,17,16,14,18);(4,6)(11,14,15)(13,17,18)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2695);