Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#11923458 |
| derived | A42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | S42 |
| commutator, hypocenter | A42 | C2×C3×C4 |
| Frattini | C2 | C3×S42 |
| Fitting, socle | C25×C3 | D32 |
| prime radical | C2×C3×A4⋊S4 | C2 |
| cosocle | C2×A42 | C22×C3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C3 | |
1124 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 108 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(8,9,11,10)(12,15,14,13);(4,5,6,7)(9,10,11)(14,15)150 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11923458, C3×#72215293, #72253015 |
| maximal normal subgroup of | D3×#11923458, C3×#72215293, #72253015 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(8,9,11,10)(12,15,14,13), (4,5,6,7)(9,10,11)(14,15));