Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | D4×B(4,𝔽2)⋊C3 |
| Fitting, Sylow | D4×B(4,𝔽2) ⋊ C3 |
2743 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 2743 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4)(6,10,7)(9,12,11);(2,4)(5,12)(6,11,9,7)(8,10);(2,4)80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#73026, #72246181, #72246216, … |
| maximal normal subgroup of | D4×#73026, #72246181, #72246216, … |
| maximal quotient of | C2×D4×B(4,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(6,10,7)(9,12,11), (2,4)(5,12)(6,11,9,7)(8,10), (2,4));