Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | Q8×#1020⋊C2 |
| Fitting | Q8×#120 ⋊ D3 |
3465 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 3465 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,3,4,8)(2,5,6,7)(10,16,14)(11,15,12);(1,2,4,6)(3,7,8,5);(9,10)(11,14)(12,13)(15,16);(12,15)(13,16)70 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×#1020⋊C2, SD16×#1020⋊C2, #41×#1020⋊C2 |
| maximal normal subgroup of | SL(2,𝔽3)×#1020⋊C2, SD16×#1020⋊C2, #41×#1020⋊C2 |
| maximal quotient of | C4⋊C4×#1020⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,4,8)(2,5,6,7)(10,16,14)(11,15,12), (1,2,4,6)(3,7,8,5), (9,10)(11,14)(12,13)(15,16), (12,15)(13,16));