Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of AΣL(2,𝔽4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C4 |
| affine special semilinear | AΣL(2,𝔽4) |
| Fitting, socle | C24 ⋊ S5 |
| derived | #11757162 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #11757162 | C2 |
| Fitting, socle | C24 | S5 |
| LCS | G ▷ #11757162 | |
| UCS | C1 | |
129 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 4 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,3,7)(2,4,8)(5,11,10)(6,12,9)(13,14,15);(1,3)(2,11,6,15)(4,13,5,9)(7,12,16,10)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 15 | 40 | 60 | 60 | 120 | 120 | 240 | 240 | 320 | 320 | 384 |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 3 | 6 | 5 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 |
| χ3 | 4 | 4 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | -1 | -1 |
| χ4 | 4 | 4 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 1 | 1 | -1 |
| χ5 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 0 |
| χ6 | 5 | 5 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 |
| χ7 | 6 | 6 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 |
| χ8 | 15 | -1 | 3 | 3 | -1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 |
| χ9 | 15 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | 1 | 0 | 0 | 0 |
| χ10 | 15 | -1 | -3 | 3 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 |
| χ11 | 15 | -1 | -3 | -1 | 3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 |
| χ12 | 30 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| maximal subgroup of | AΓL(2,𝔽4), #72218848, #72218850, … |
| maximal normal subgroup of | #72218852, C3×AΣL(2,𝔽4), C2×AΣL(2,𝔽4) |
| maximal quotient of | C3×AΣL(2,𝔽4), #72241553, C2×AΣL(2,𝔽4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240993);