Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C8×PSU(3,𝔽2)⋊C3 |
| Fitting | C2×C32×C8 ⋊ SL(2,𝔽3) |
| derived | PSU(3,𝔽2) ⋊ C2×C3×C8 |
408 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 71 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 — 2 generators on 19 points:
(1,2,4)(3,8,9)(5,7,6)(12,13,14,16,15,17,18,19);(2,8,6,3)(4,5,7,9)(10,11)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×AGL(2,𝔽3), #97×PSU(3,𝔽2)⋊C3, #72244970, … |
| maximal normal subgroup of | C2×C8×AGL(2,𝔽3), #97×PSU(3,𝔽2)⋊C3, #72244970, … |
| maximal quotient of | M32×PSU(3,𝔽2)⋊C3 |
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)(12,13,14,16,15,17,18,19), (2,8,6,3)(4,5,7,9)(10,11));