Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#13181779 |
| Fitting, socle | C23×C32 ⋊ #214 |
912 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 15 points:
(3,6,5)(7,10,14,12,13,11)(8,15,9);(1,2);(4,6)(8,9,11,12)(10,14,15,13)54 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×AΓL(1,𝔽9) |
| maximal normal subgroup of | C2×S4×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5)(7,10,14,12,13,11)(8,15,9), (1,2), (4,6)(8,9,11,12)(10,14,15,13));