Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D7×AGL(2,𝔽3) |
| Fitting, socle | C32×C7 ⋊ C2×GL(2,𝔽3) |
| derived | C7×PSU(3,𝔽2)⋊C3 ⋊ C22 |
| Sylow | C7 ⋊ C2×AGL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×PSU(3,𝔽2)⋊C3 | C22 |
| Fitting, socle | C32×C7 | C2×GL(2,𝔽3) |
| LCS | G ▷ C7×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 | |
288 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,7,4)(2,6,3)(5,9,8)(11,12)(13,14)(15,16);(2,4)(3,6)(7,8)(10,11,13,15,16,14,12)55 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(2,𝔽3), C2×D7×AGL(2,𝔽3) |
| maximal normal subgroup of | AGL(1,𝔽7)×AGL(2,𝔽3), C2×D7×AGL(2,𝔽3) |
| maximal quotient of | C2×D7×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,4)(2,6,3)(5,9,8)(11,12)(13,14)(15,16), (2,4)(3,6)(7,8)(10,11,13,15,16,14,12));