Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C34 |
| direct product | C3×D8×H(3,𝔽3) |
| unambiguous semidirect product | (C32×D8)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C32×D4 |
| commutator, Frattini, cosocle | C3×C4 | C22×C33 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C3×C4×H(3,𝔽3) ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(13,14,15,16,17,18,19,20);(4,5,7)(6,9,11)(8,12,10)(14,20)(15,19)(16,18);(5,8,9)(7,11,12)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1768);