Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C3 |
| direct product | #97×PSU(3,𝔽2)⋊C3 |
| Fitting | C32×#97 ⋊ SL(2,𝔽3) |
814 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 288 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4)(3,8,9)(5,7,6)(14,15,16,19,17,18,20,21);(2,8,6,3)(4,5,7,9)(10,11,12,13)(15,18)(19,21)200 conjugacy classes — view the full character table.
| maximal subgroup of | #97×AGL(2,𝔽3) |
| maximal normal subgroup of | #97×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4)(3,8,9)(5,7,6)(14,15,16,19,17,18,20,21), (2,8,6,3)(4,5,7,9)(10,11,12,13)(15,18)(19,21));