Finite Group DatabasebetaFinite Group Databasebeta| order | 21600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3 ⋊ #72219548 |
| socle, derived | C3×A52 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×A52 | C2 |
| Fitting | C3 | #72219548 |
| LCS | G ▷ C3×A52 | |
| UCS | C1 | |
650 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 75 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3)(4,6,5)(9,12)(10,13);(1,2)(4,6,7)(5,8)(9,11,13)(10,12)51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72219548 |
| maximal normal subgroup of | D3×#72219548 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,6,5)(9,12)(10,13), (1,2)(4,6,7)(5,8)(9,11,13)(10,12));