Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#11724027 |
| derived | (C22⋊A4)⋊C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11724027 |
| commutator, cosocle, hypocenter | (C22⋊A4)⋊C9 | C2×C3 |
| Frattini | C3 | C3×A4⋊S4 |
| Fitting, socle | C24×C32 | C3⋊D3 |
| LCS | G ▷ (C22⋊A4)⋊C9 | |
| UCS | C1 ◁ C3 | |
411 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(5,7)(6,10)(8,11)(9,12)(13,16,15,14)(17,20,18,19);(4,9,7,6,11,8,10,5,12)(18,19,20);(14,15,16)(18,19,20)90 conjugacy classes — view the full character table.
| maximal subgroup of | #72259316, A4×#11724027, C3×#72213207, … |
| maximal normal subgroup of | #72259316, C3×#72213207, C2×C3×#11724027 |
| maximal quotient of | #72229343, A4×#11724027, C2×C3×#11724027 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,7)(6,10)(8,11)(9,12)(13,16,15,14)(17,20,18,19), (4,9,7,6,11,8,10,5,12)(18,19,20), (14,15,16)(18,19,20));