Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#13181780 |
| Fitting, socle | C23×C32 ⋊ #215 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #13181780 |
| commutator, cosocle | A4×#157 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | #215 |
| prime radical | C2×#157⋊S4 | C2 |
| hypocenter | C32×A4 | C2×SD16 |
| LCS | G ▷ A4×#157 ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
821 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 821 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)(7,15,12,8)(9,13,11,14);(1,2);(4,6)(8,10)(9,14)(12,13)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213290, 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)(7,15,12,8)(9,13,11,14), (1,2), (4,6)(8,10)(9,14)(12,13));