Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#85554 |
| Fitting | C5×Q82 ⋊ D3 |
| derived | Q8⋊SL(2,𝔽3) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #5934 |
| commutator, cosocle, hypocenter | Q8⋊SL(2,𝔽3) | C2×C5 |
| Frattini | C22 | C5×#1020 |
| Fitting | C5×Q82 | D3 |
| hypercenter | C22×C5 | #1020 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
218 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,16,18,7,11,13,8,15)(9,21,17,19,20,14,10,12);(8,9,10)(12,13,14)(16,21,19)(17,18,20)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236990);