Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | D3×#5450⋊C3 |
| Fitting, derived | C3×H(3,𝔽4) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×H(3,𝔽4) | C2×C32 |
| Frattini | C22 | D3×A42 |
| socle | C22×C3 | C2×A42 |
| LCS | G ▷ C3×H(3,𝔽4) | |
| UCS | C1 | |
548 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,6)(2,3,16,5,9,4)(7,15)(8,13,14,12,10,11)(18,19);(1,8)(2,3)(4,6,16,15,12,7)(5,11,13,9,10,14)(17,18,19)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(2,𝔽4), D3×#11921467 |
| maximal normal subgroup of | D3×#11921467 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,6)(2,3,16,5,9,4)(7,15)(8,13,14,12,10,11)(18,19), (1,8)(2,3)(4,6,16,15,12,7)(5,11,13,9,10,14)(17,18,19));