Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×D3×PSU(3,𝔽2)⋊C3 |
| Fitting, socle | C22×C33 ⋊ C2×SL(2,𝔽3) |
| derived | C3×PSU(3,𝔽2) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | C3×PSU(3,𝔽2) | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C2×SL(2,𝔽3) |
| LCS | G ▷ C3×PSU(3,𝔽2) | |
| UCS | C1 ◁ C22 | |
1594 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 16 points:
(1,2,3)(4,5,9)(6,10,8)(7,11,12)(15,16);(2,3)(5,11,10,7)(6,12,9,8);(13,14)120 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3⋊D3×PSU(3,𝔽2)⋊C3, C2×D32×PSU(3,𝔽2)⋊C3, C22×D3×AGL(2,𝔽3) |
| maximal normal subgroup of | C22×D3×AGL(2,𝔽3) |
| maximal quotient of | C22×C3⋊D3×PSU(3,𝔽2)⋊C3, C2×D32×PSU(3,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,9)(6,10,8)(7,11,12)(15,16), (2,3)(5,11,10,7)(6,12,9,8), (13,14));