Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#85555 |
| Fitting | C5×Q82 ⋊ D3 |
| derived | Q8⋊SL(2,𝔽3) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C5 | #1020 |
| commutator, cosocle, hypocenter | Q8⋊SL(2,𝔽3) | C2×C5 |
| Frattini | C22 | C5×#1020 |
| Fitting | C5×Q82 | D3 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C5 | |
238 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 320 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,12,11,8,9,7,13,10)(14,20,19,16,17,15,21,18);(7,8,13)(10,11,12)(15,19,16)(18,20,21)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236991);