Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×D7×PSU(3,𝔽2)⋊C3 |
| Fitting, socle | C2×C32×C7 ⋊ C2×SL(2,𝔽3) |
| derived | C7×PSU(3,𝔽2) ⋊ C22×C3 |
| Sylow | C7 ⋊ C22×PSU(3,𝔽2)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | C7×PSU(3,𝔽2) | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C2×SL(2,𝔽3) |
| LCS | G ▷ C7×PSU(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
328 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,6)(3,8,4)(5,9,7)(11,12)(13,14)(15,16);(2,3,7,4)(5,9,8,6)(10,11,13,15,16,14,12)(17,18)100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×PSU(3,𝔽2)⋊C3, C2×D7×AGL(2,𝔽3) |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×PSU(3,𝔽2)⋊C3, C2×D7×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,6)(3,8,4)(5,9,7)(11,12)(13,14)(15,16), (2,3,7,4)(5,9,8,6)(10,11,13,15,16,14,12)(17,18));