Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C37 |
| direct product | C3×H(3,𝔽3)×AGL(1,ℤ/9) |
| Fitting, Sylow | C3×H(3,𝔽3)×3−1+2 ⋊ C2 |
6028 conjugacy classes of subgroups; only the 734 normal subgroups are shown.
This group has 6028 conjugacy classes of subgroups — too many to draw. Only its 734 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) = 162 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3)(14,16)(15,19)(17,20)(18,21);(5,8,9)(7,11,12)(13,17,16,15,21,18,19,14,20);(14,17,21)(16,20,18);(4,5,7)(6,9,11)(8,12,10)(14,17,21)(16,20,18)330 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72243853, D3×H(3,𝔽3)×AGL(1,ℤ/9), C3×(C3⋊D3)⋊C3×AGL(1,ℤ/9), … |
| maximal normal subgroup of | #72243853, D3×H(3,𝔽3)×AGL(1,ℤ/9), C3×(C3⋊D3)⋊C3×AGL(1,ℤ/9), … |
| maximal quotient of | #72243853 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(14,16)(15,19)(17,20)(18,21), (5,8,9)(7,11,12)(13,17,16,15,21,18,19,14,20), (14,17,21)(16,20,18), (4,5,7)(6,9,11)(8,12,10)(14,17,21)(16,20,18));