Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C33 |
| direct product | C22×C3⋊(A4⋊S4) |
| Fitting, socle | C26×C3 ⋊ C3⋊D3 |
| derived | C3×A42 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(A4⋊S4) |
| commutator, cosocle, hypocenter | C3×A42 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C26×C3 | C3⋊D3 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 ◁ C22 | |
4972 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 4972 conjugacy classes of subgroups — too many to draw. Only its 221 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 — 4 generators on 15 points:
(3,4)(5,6)(7,8)(9,12)(10,11)(13,14,15);(6,7)(11,12)(14,15);(1,2)(6,7,8)(10,11,12);(10,12,11)120 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D3×A4⋊S4, C22×#72213301, #72252927 |
| maximal normal subgroup of | C22×D3×A4⋊S4, C22×#72213301, #72252927 |
| maximal quotient of | C22×#72213251 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,6)(7,8)(9,12)(10,11)(13,14,15), (6,7)(11,12)(14,15), (1,2)(6,7,8)(10,11,12), (10,12,11));