Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×AΣL(2,𝔽4) |
| Fitting, socle | C25 ⋊ S5 |
| derived | #11757162 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AΣL(2,𝔽4) |
| commutator, cosocle, hypocenter | #11757162 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C25 | S5 |
| LCS | G ▷ #11757162 | |
| UCS | C1 ◁ C2 | |
534 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,10,16,17)(4,6,11,14)(5,8,18,13)(7,15,12,9);(4,10)(5,11,9,13)(6,16,7,17)(12,15,18,14)24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AΓL(2,𝔽4), D3×AΣL(2,𝔽4), C2×#72218848, … |
| maximal normal subgroup of | C2×#72218852, C4×AΣL(2,𝔽4) |
| maximal quotient of | D3×AΣL(2,𝔽4), C4×AΣL(2,𝔽4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,10,16,17)(4,6,11,14)(5,8,18,13)(7,15,12,9), (4,10)(5,11,9,13)(6,16,7,17)(12,15,18,14));