Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#5450⋊C4 |
1859 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 1859 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 22 points:
(7,9,21,10,17,20)(8,13,12)(11,15,14,18,16,22);(3,4,5,6)(8,9)(10,11)(12,17)(13,16)(15,19)(20,21);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72258617 |
| maximal normal subgroup of | C2×#72258617 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,9,21,10,17,20)(8,13,12)(11,15,14,18,16,22), (3,4,5,6)(8,9)(10,11)(12,17)(13,16)(15,19)(20,21), (1,2));