Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#11921467 |
| Fitting | C3×H(3,𝔽4) ⋊ C3⋊D3 |
| derived | #5450⋊C3 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11921467 |
| commutator, cosocle, hypocenter | #5450⋊C3 | C2×C3 |
| Frattini | C22 | C3×A4⋊S4 |
| Fitting | C3×H(3,𝔽4) | C3⋊D3 |
| socle | C22×C3 | A4⋊S4 |
| LCS | G ▷ #5450⋊C3 | |
| UCS | C1 ◁ C3 | |
456 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 3 generators on 19 points:
(1,8,7,3,4,10,16,12)(2,14,15,13,9,11,6,5)(17,18,19);(2,6,14)(3,11,8)(4,13,5)(7,9,12)(10,15,16);(2,3,7)(4,5,13)(6,11,9)(8,12,14)(10,15,16)51 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AΓL(2,𝔽4), D3×#11921467, C3×#72215868, … |
| maximal normal subgroup of | D3×#11921467, C2×C3×#11921467 |
| maximal quotient of | C2×C3×#11921467 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,7,3,4,10,16,12)(2,14,15,13,9,11,6,5)(17,18,19), (2,6,14)(3,11,8)(4,13,5)(7,9,12)(10,15,16), (2,3,7)(4,5,13)(6,11,9)(8,12,14)(10,15,16));