Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | D3×SL(2,𝔽3)×S4 |
| Fitting | C22×C3×Q8 ⋊ C2×C3×D3 |
| derived | C3×Q8×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A4×S4 |
| commutator, cosocle, hypocenter | C3×Q8×A4 | C22×C3 |
| Fitting | C22×C3×Q8 | C2×C3×D3 |
| socle | C23×C3 | C2×D3×A4 |
| LCS | G ▷ C3×Q8×A4 | |
| UCS | C1 ◁ C2 | |
1215 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4)(6,7)(9,10,14)(12,13,15);(2,3)(5,6,7)(8,12,11,14)(9,15,13,10)105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×S4×GL(2,𝔽3) |
| maximal normal subgroup of | D3×S4×GL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4)(6,7)(9,10,14)(12,13,15), (2,3)(5,6,7)(8,12,11,14)(9,15,13,10));