Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×#72220166 |
| derived | C22×C22⋊C9 ⋊ C22×C9 |
1222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(4,5,6)(7,8,10)(11,15,19,13,18,14,16,12,17)(20,27,26,23,24,28,21,22,25);(3,6)(4,5)(8,9)(21,23)(22,25)(24,26)(27,28)168 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220173 |
| maximal normal subgroup of | C2×#72220173 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,6)(7,8,10)(11,15,19,13,18,14,16,12,17)(20,27,26,23,24,28,21,22,25), (3,6)(4,5)(8,9)(21,23)(22,25)(24,26)(27,28));