Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×D8×PSU(3,𝔽2)⋊C3 |
| Fitting | C2×C32×D8 ⋊ SL(2,𝔽3) |
1866 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4)(3,8,9)(5,7,6)(12,13,17,19,15,18,14,16);(2,8,6,3)(4,5,7,9)(10,11);(13,16)(14,17)(18,19)140 conjugacy classes — view the full character table.
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)(12,13,17,19,15,18,14,16), (2,8,6,3)(4,5,7,9)(10,11), (13,16)(14,17)(18,19));