Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C22×#11724029 |
| derived | (C3×A4)⋊A4 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | A4⋊S4 |
| commutator, cosocle, hypocenter | (C3×A4)⋊A4 | C23 |
| Frattini | C3 | C22×A4⋊S4 |
| Fitting, socle | C26×C3 | C3⋊D3 |
| LCS | G ▷ (C3×A4)⋊A4 | |
| UCS | C1 ◁ C22×C3 | |
3370 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 3370 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,5,6)(4,8,9)(12,13)(14,17)(15,16)(18,19,21);(2,4)(5,9)(6,8)(15,17)(19,20);(1,2,4)(3,6,8)(5,9,7)(10,11)(15,16,17)(19,20,21)136 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72213251 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5,6)(4,8,9)(12,13)(14,17)(15,16)(18,19,21), (2,4)(5,9)(6,8)(15,17)(19,20), (1,2,4)(3,6,8)(5,9,7)(10,11)(15,16,17)(19,20,21));