Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#3026⋊C3 |
| unambiguous semidirect product | #3026⋊(C3×C5) |
| unambiguous semidirect product | (C5×#3026)⋊C3 |
| Fitting, Hall | C5×#3026 ⋊ C3 |
| Sylow | #3026 ⋊ 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×#3026 | C3 |
| hypercenter | C22×C5 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
360 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 360 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)(7,11)(8,15,18)(9,16,21)(12,20,17,14,19,13);(6,20,10,19)(7,21,11,15)(8,13)(9,12)(14,18)(16,17)115 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#72209831, D5×#3026⋊C3 |
| maximal normal subgroup of | D5×#3026⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236682);