Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#85565 |
| derived | #997 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C5 | #1020 |
| commutator, hypocenter | #997 | C4×C5 |
| Frattini | C22 | C5×#1020 |
| Fitting | C2×C5×H(5,𝔽2) | D3 |
| prime radical | C2×C5×#997 | C2 |
| cosocle | C2×#997 | C2×C5 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C5 | |
398 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4,5)(6,7,8,9)(10,11,15,12,13,16,17,14);(11,15,12)(14,16,17)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237001);