Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C211 |
| direct product | C22×ΓL(1,𝔽8)×#120 |
| Fitting, Hall | C22×C7×#120 ⋊ C3 |
| Sylow | C7 ⋊ C22×C3×#120 |
9088 conjugacy classes of subgroups; only the 1668 normal subgroups are shown.
This group has 9088 conjugacy classes of subgroups — too many to draw. Only its 1668 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) = 224 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(14,15,16)(17,19,18);(3,4)(13,14,16,18,15,17,19);(6,9)(10,12);(6,10)(9,12);(5,6)(7,9)(8,10)(11,12)280 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#120×AGL(1,𝔽7), ΓL(1,𝔽8)×#12687, #72276646, … |
| maximal normal subgroup of | C22×#120×AGL(1,𝔽7), ΓL(1,𝔽8)×#12687, #72276646, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(14,15,16)(17,19,18), (3,4)(13,14,16,18,15,17,19), (6,9)(10,12), (6,10)(9,12), (5,6)(7,9)(8,10)(11,12));