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) = C32 |
| direct product | SD16×C32⋊C9 |
| unambiguous semidirect product | (C32×SD16)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C32×D4 |
| commutator | C3×C4 | C22×C3×C9 |
| Frattini, cosocle | C32×C4 | C22×C32 |
| prime radical | C33×D4 | C2×C3 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C4×C32⋊C9 ◁ G | |
230 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7,8)(9,10,12)(11,14,16)(13,17,15)(18,19,21,20,22,24,23,25,26);(2,4)(3,7)(6,8)(10,13,14)(12,16,17)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 613);