Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#1020⋊A4 |
| Fitting, socle | C26×C3 ⋊ C3×D3 |
| derived | C22×C22⋊A4 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #1020⋊A4 |
| commutator, cosocle, hypocenter | C22×C22⋊A4 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C26×C3 | C3×D3 |
| LCS | G ▷ C22×C22⋊A4 | |
| UCS | C1 ◁ C3 | |
1275 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 1275 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 36 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,4,3)(5,12,9,11,7,10)(6,8);(1,4)(2,3)(7,9,8)(10,11,12)(13,14,15)72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72215854, C2×C3×#1020⋊A4, C3×#72215889 |
| maximal normal subgroup of | C2×C3×#1020⋊A4, C3×#72215889 |
| maximal quotient of | C3×#72215845, C2×C3×#1020⋊A4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(5,12,9,11,7,10)(6,8), (1,4)(2,3)(7,9,8)(10,11,12)(13,14,15));