Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×Q8×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C32 | C23×C32 |
| prime radical | C22×AΣL(1,𝔽27) | C22 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C32×Q8 ◁ G | |
380 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(11,12,14)(13,15,18)(16,19,17);(3,4,5,6)(7,10,9,8)(12,15,16);(3,7,5,9)(4,8,6,10)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1224);