Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D4 ⋊ SO(3,𝔽7) |
232 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,5)(4,6)(7,8)(10,13,15,16,12,14,11);(1,4)(2,6)(3,7)(5,8)(9,16,14,11,10,15)33 conjugacy classes — view the full character table.
| maximal subgroup of | #72244493 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,5)(4,6)(7,8)(10,13,15,16,12,14,11), (1,4)(2,6)(3,7)(5,8)(9,16,14,11,10,15));