Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C4⋊C4×AΣL(1,𝔽27) |
| unambiguous semidirect product | C4⋊(C4×AΣL(1,𝔽27)) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×H(3,𝔽3) |
| commutator | C2×C32 | C2×C32×C4 |
| Frattini, cosocle | C22×C32 | C22×C32 |
| prime radical | C22×AΣL(1,𝔽27) | C22 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C32×C4⋊C4 ◁ G | |
260 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 144 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,4)(5,6,7,8)(9,10,12)(11,13,16)(14,17,15);(1,2,3,4)(10,13,14)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 581);