Finite Group DatabasebetaFinite Group Databasebeta| order | 3360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | D7×A4⋊AGL(1,𝔽5) |
| Fitting, socle | C22×C5×C7 ⋊ C2×Dic3 |
| derived | C5×C7×A4 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C2×SL(2,ℤ/4) |
| Sylow | C7 ⋊ C2×A4⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ D7×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×C7×A4 | C2×C4 |
| Fitting, socle | C22×C5×C7 | C2×Dic3 |
| prime radical | D5×A4×D7 | C2 |
| cosocle | C7×D5×A4 | C22 |
| LCS | G ▷ C5×C7×A4 | |
| UCS | C1 | |
252 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 140 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,4,3)(6,7)(8,9)(10,11)(12,13,16,14,15);(2,4)(5,6,8,10,11,9,7)(13,14,15,16)70 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×A4⋊AGL(1,𝔽5), D7×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | AGL(1,𝔽7)×A4⋊AGL(1,𝔽5), D7×AGL(1,𝔽5)×S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7)(8,9)(10,11)(12,13,16,14,15), (2,4)(5,6,8,10,11,9,7)(13,14,15,16));