Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#73028 |
| Fitting | C5×B(4,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C5 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C22×C5 |
| Frattini | C23 | C2×C5×S4 |
| Fitting | C5×B(4,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C5 | |
256 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 22 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)(6,15,16)(7,12,20,11,17,21)(8,19,10)(9,18);(6,7)(8,14)(9,17)(10,11)(12,18)(13,15)(16,21)(19,20)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236450);