Finite Group DatabasebetaFinite Group Databasebeta| order | 17280 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, socle | C24×C3 ⋊ ΓL(2,𝔽4) |
| derived | C3×AGL(2,𝔽4) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AGL(2,𝔽4) | C2 |
| Fitting, socle | C24×C3 | ΓL(2,𝔽4) |
| LCS | G ▷ C3×AGL(2,𝔽4) | |
| UCS | C1 | |
660 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 15 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 — 3 generators on 19 points:
(2,3)(4,6,10,15,8,13)(5,14,17,18,9,16)(7,12,19);(2,3)(5,13)(6,9)(7,18)(10,11)(12,19)(14,16);(1,2,3)(4,8,14,5,18,17)(6,11,15)(7,9,10,16,12,19)36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AΓL(2,𝔽4) |
| maximal normal subgroup of | D3×AΓL(2,𝔽4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,6,10,15,8,13)(5,14,17,18,9,16)(7,12,19), (2,3)(5,13)(6,9)(7,18)(10,11)(12,19)(14,16), (1,2,3)(4,8,14,5,18,17)(6,11,15)(7,9,10,16,12,19));