Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D21×B(4,𝔽2) |
| Fitting | C3×C7×B(4,𝔽2) ⋊ C2 |
| Hall | C3×C7 ⋊ C2×B(4,𝔽2) |
| Sylow | C7 ⋊ D3×B(4,𝔽2) |
| Sylow | C3 ⋊ D7×B(4,𝔽2) |
2036 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 2036 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,4,6,7,5,3)(8,9,10)(11,13)(12,15)(14,17)(16,18);(2,3)(4,5)(6,7)(9,10);(15,18);(11,12,14,16)(13,15,17,18)192 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×B(4,𝔽2), B(4,𝔽2)×C3⋊AGL(1,𝔽7) |
| maximal normal subgroup of | D3×D7×B(4,𝔽2), B(4,𝔽2)×C3⋊AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,7,5,3)(8,9,10)(11,13)(12,15)(14,17)(16,18), (2,3)(4,5)(6,7)(9,10), (15,18), (11,12,14,16)(13,15,17,18));