Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | A4×C3⋊#1020 |
| Fitting, socle | C26×C3 ⋊ C3×D3 |
| derived | C22×C3×C22⋊A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C3×C22⋊A4 | C2×C3 |
| Fitting, socle | C26×C3 | C3×D3 |
| LCS | G ▷ C22×C3×C22⋊A4 | |
| UCS | C1 | |
1660 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 1660 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 3 generators on 15 points:
(2,3,4)(5,8,7,6)(9,12,10,11)(14,15);(1,4)(2,3)(13,14,15);(6,7,8)(10,11,12)72 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#11724027, #72269067, C3×A4×C3⋊#1020 |
| maximal normal subgroup of | C3×A4×C3⋊#1020 |
| maximal quotient of | C3×A4×C3⋊#1020 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,8,7,6)(9,12,10,11)(14,15), (1,4)(2,3)(13,14,15), (6,7,8)(10,11,12));