Finite Group DatabasebetaFinite Group Databasebeta| order | 10368 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×AGL(2,𝔽3) |
| Fitting, socle | C22×C33 ⋊ C2×GL(2,𝔽3) |
| derived | C3×PSU(3,𝔽2)⋊C3 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×AGL(2,𝔽3) |
| commutator, cosocle, hypocenter | C3×PSU(3,𝔽2)⋊C3 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C2×GL(2,𝔽3) |
| LCS | G ▷ C3×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 ◁ C22 | |
7077 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 7077 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 16 points:
(6,7)(8,10,15)(9,11,14)(12,13,16);(5,6,7)(9,11)(10,14)(13,15);(3,4);(1,2)132 conjugacy classes — view the full character table.
| automorphism group of | D3×C3⋊Dic3, Dic3×C3⋊D3, #6289, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7)(8,10,15)(9,11,14)(12,13,16), (5,6,7)(9,11)(10,14)(13,15), (3,4), (1,2));