Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#3027⋊C3 |
| unambiguous semidirect product | #3027⋊(C3×C5) |
| unambiguous semidirect product | (C5×#3027)⋊C3 |
| Fitting, Hall | C5×#3027 ⋊ C3 |
| Sylow | #3027 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3×C5 |
| Frattini | C23 | C22×C5×A4 |
| Fitting | C5×#3027 | C3 |
| hypercenter | C4×C5 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
344 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 36 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)(6,7,10,11)(8,20,9,18,15,19,14,21,17,12,16,13);(6,15,10,16)(7,21,11,20)(8,9)(12,19)(13,18)(14,17)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236683);