Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C3 |
| direct product | SD16×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D4×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32×C4 | C22×C32 |
| prime radical | D4×AΣL(1,𝔽27) | C2 |
| LCS | G ▷ C32×C4 ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C32×C4 ◁ G | |
200 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4,5,6,7,8)(9,10,12)(11,13,16)(14,17,15);(2,4)(3,7)(6,8)(10,13,14)119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 617);