Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D3×(#1020⋊C3)⋊C2 |
| Fitting, socle | C24×C3 ⋊ C2×D32 |
| derived | C3×A42 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A42 | C23 |
| Fitting, socle | C24×C3 | C2×D32 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 | |
1144 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,3,5,4)(2,6)(9,10);(1,2)(3,7,4,8,5,6);(10,11)48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AΓL(2,𝔽4), D3×#10596770⋊C2, C2×D3×(#1020⋊C3)⋊C2 |
| maximal normal subgroup of | D3×#10596770⋊C2, C2×D3×(#1020⋊C3)⋊C2 |
| maximal quotient of | C2×D3×(#1020⋊C3)⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,5,4)(2,6)(9,10), (1,2)(3,7,4,8,5,6), (10,11));