Finite Group DatabasebetaFinite Group Databasebeta| order | 8640 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×AGL(2,𝔽4) |
| Fitting, socle | C24×C3 ⋊ C3×A5 |
| derived | #11757162 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(2,𝔽4) |
| commutator, hypocenter | #11757162 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C3×A5 |
| cosocle | C24 | C32×A5 |
| LCS | G ▷ #11757162 | |
| UCS | C1 ◁ C3 | |
291 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(4,16,13,18,5,9,14,8,11,17,6,19,7,12,15);(1,2,3)(4,8,14)(6,15,13)(9,19,17)(10,12,11)57 conjugacy classes — view the full character table.
| derived subgroup of | #72241556, D3×AΓL(2,𝔽4) |
| maximal subgroup of | C3×AΓL(2,𝔽4), #72241556, D3×AGL(2,𝔽4) |
| maximal normal subgroup of | C3×AΓL(2,𝔽4), #72241556, D3×AGL(2,𝔽4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,16,13,18,5,9,14,8,11,17,6,19,7,12,15), (1,2,3)(4,8,14)(6,15,13)(9,19,17)(10,12,11));