Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | AGL(1,𝔽7)×AΓL(1,𝔽9) |
| Fitting, socle | C32×C7 ⋊ C2×C3×SD16 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2×SD16 |
| Sylow | C7 ⋊ C2×C3×AΓL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C7×#157 | C23×C3 |
| Fitting, socle, hypocenter | C32×C7 | C2×C3×SD16 |
| prime radical | AGL(1,𝔽7)×GO+(4,𝔽2) | C2 |
| LCS | G ▷ C7×#157 ▷ C7×C3⋊D3 ▷ C32×C7 | |
| UCS | C1 | |
356 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 16 points:
(1,7,5,8)(2,4,3,6)(10,11,16)(13,14,15);(11,12)(13,14)(15,16);(2,4)(3,5)(8,9)63 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(2,𝔽3), C2×AGL(1,𝔽7)×AΓL(1,𝔽9) |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×AΓL(1,𝔽9) |
| maximal quotient of | C2×AGL(1,𝔽7)×AΓL(1,𝔽9) |
| automorphism group of | C7⋊#157 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,5,8)(2,4,3,6)(10,11,16)(13,14,15), (11,12)(13,14)(15,16), (2,4)(3,5)(8,9));