Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C32 |
| direct product | C3×M16×3−1+2 |
| unambiguous semidirect product | (C9×M16)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22×C32 |
| commutator | C2×C3 | C2×C33×C4 |
| Frattini, cosocle | C3×C4 | C22×C33 |
| socle | C2×C32 | C2×C32×C4 |
| prime radical | C22×C33 | C3×C4 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
380 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 288 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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);(4,5,7,6,9,11,10,8,12)(14,18)(16,20);(5,8,9)(7,11,12)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1765);