Finite Group DatabasebetaFinite Group Databasebeta| order | 11520 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | AGL(2,𝔽4) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AΓL(2,𝔽4) |
| commutator, hypocenter | AGL(2,𝔽4) | C4 |
| Fitting, socle | C25 | ΓL(2,𝔽4) |
| prime radical, cosocle | C2×AGL(2,𝔽4) | C2 |
| LCS | G ▷ AGL(2,𝔽4) | |
| UCS | C1 ◁ C2 | |
399 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,3,2,4)(5,16)(6,9)(7,13)(10,14)(12,19)(17,18);(1,3,2,4)(5,16,13,20,12,6,19,14)(7,11,17,8,9,10,15,18)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×AΓL(2,𝔽4) |
| maximal normal subgroup of | C4×AΓL(2,𝔽4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,4)(5,16)(6,9)(7,13)(10,14)(12,19)(17,18), (1,3,2,4)(5,16,13,20,12,6,19,14)(7,11,17,8,9,10,15,18));