Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×A4⋊AGL(1,𝔽5) |
| Fitting, socle | C24×C5 ⋊ C3×Dic3 |
| derived | C22×C5×A4 ⋊ C3×C4 |
| Sylow | C5 ⋊ A4×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C22×C5×A4 | C3×C4 |
| Fitting, socle | C24×C5 | C3×Dic3 |
| prime radical | D5×A42 | C2 |
| cosocle | C22×D5×A4 | C2×C3 |
| LCS | G ▷ C22×C5×A4 | |
| UCS | C1 | |
334 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,4,3)(5,8,6,7)(10,11,12,13);(1,4)(2,3)(6,7,8)(9,10,13,11,12)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×AGL(1,𝔽5)×S4, S4×A4⋊AGL(1,𝔽5), #72231215, … |
| maximal normal subgroup of | A4×AGL(1,𝔽5)×S4, S4×A4⋊AGL(1,𝔽5), #72231215, … |
| maximal quotient of | A4×A4⋊#1044, #72277301, C2×A4×A4⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(5,8,6,7)(10,11,12,13), (1,4)(2,3)(6,7,8)(9,10,13,11,12));