Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×D4×C9⋊C9 |
| unambiguous semidirect product | C4⋊(C22×C9⋊C9) |
| unambiguous semidirect product | (C2×D4×C9)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×C32 |
| commutator | C2×C3 | C23×C3×C9 |
| Frattini, cosocle | C2×C32 | C23×C32 |
| prime radical | C2×C32×D4 | C32 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
459 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(8,10,13)(11,15,14)(16,17,19,18,20,22,21,23,24);(3,4,5,6)(7,8,11,9,10,14,12,13,15);(4,6)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1212);