Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #3532⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3532⋊C3 |
| unambiguous semidirect product | #3532⋊(C3×C5) |
| unambiguous semidirect product | (C5×#3532)⋊C3 |
| Fitting, Hall | C5×#3532 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C2×C3×C5 |
| Sylow | #3532 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #3532⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C3×C5 |
| Frattini | C22 | C2×C5×C22⋊A4 |
| Fitting | C5×#3532 | C3 |
| socle | C22×C5 | C2×C22⋊A4 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C5 | |
532 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 3 generators on 21 points:
(1,2,3,4,5)(7,19,20,12,13,21)(8,14,16,17,18,15)(9,10,11);(7,15)(8,13)(12,16)(14,18)(17,19)(20,21);(6,20)(7,13)(8,12)(9,14)(10,18)(11,21)(15,19)(16,17)90 conjugacy classes — view the full character table.
| derived subgroup of | #72243488, #72243493, #72243498, … |
| maximal subgroup of | D5×#3532⋊C3, C5×#11714270, #72221396, … |
| maximal normal subgroup of | D5×#3532⋊C3, C5×#11714270, #72221396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237095);