metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D20, D20⋊5C2, C4○Dic10, C4.16D10, Dic10⋊5C2, C10.4C23, C22.2D10, C20.16C22, D10.1C22, Dic5.2C22, (C2×C4)⋊3D5, (C2×C20)⋊4C2, (C4×D5)⋊4C2, C5⋊1(C4○D4), C4○(C5⋊D4), C5⋊D4⋊3C2, C2.5(C22×D5), (C2×C10).11C22, SmallGroup(80,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D20
G = < a,b,c | a4=c2=1, b10=a2, ab=ba, ac=ca, cbc=a2b9 >
Character table of C4○D20
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | ζ54-ζ5 | 1+√5/2 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | -ζ53+ζ52 | 1-√5/2 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | ζ53-ζ52 | 1-√5/2 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | ζ54-ζ5 | 1+√5/2 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | -ζ54+ζ5 | 1+√5/2 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | -ζ54+ζ5 | 1+√5/2 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | ζ53-ζ52 | 1-√5/2 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | -ζ53+ζ52 | 1-√5/2 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | complex faithful |
►On 40 points
Generators in S40
(1 32 11 22)(2 33 12 23)(3 34 13 24)(4 35 14 25)(5 36 15 26)(6 37 16 27)(7 38 17 28)(8 39 18 29)(9 40 19 30)(10 21 20 31)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)
G:=sub<Sym(40)| (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,36,15,26)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,21,20,31), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)>;
G:=Group( (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,36,15,26)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,21,20,31), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40) );
G=PermutationGroup([(1,32,11,22),(2,33,12,23),(3,34,13,24),(4,35,14,25),(5,36,15,26),(6,37,16,27),(7,38,17,28),(8,39,18,29),(9,40,19,30),(10,21,20,31)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40)])
C4○D20 is a maximal subgroup of
D20⋊4C4 D20⋊7C4 D20.3C4 D40⋊7C2 D20.2C4 C8⋊D10 C8.D10 D4.D10 C20.C23 D4.8D10 D4⋊6D10 Q8.10D10 D5×C4○D4 D4⋊8D10 D4.10D10 D20⋊5S3 D60⋊C2 D6.D10 Dic5.D6 D60⋊11C2 D100⋊5C2 D20⋊5D5 D10.9D10 Dic10⋊5D5 Dic5.D10 C20.50D10
C4○D20 is a maximal quotient of
C4×Dic10 C20.6Q8 C42⋊D5 C4×D20 C4.D20 C42⋊2D5 C23.D10 D10.12D4 D10⋊D4 Dic5.5D4 Dic5.Q8 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 C20.48D4 C23.21D10 C4×C5⋊D4 C23.23D10 C20⋊7D4 D20⋊5S3 D60⋊C2 D6.D10 Dic5.D6 D60⋊11C2 D100⋊5C2 D20⋊5D5 D10.9D10 Dic10⋊5D5 Dic5.D10 C20.50D10
Matrix representation of C4○D20 ►in GL2(𝔽41) generated by
| 9 | 0 |
| 0 | 9 |
| 16 | 2 |
| 39 | 28 |
| 35 | 1 |
| 6 | 6 |
G:=sub<GL(2,GF(41))| [9,0,0,9],[16,39,2,28],[35,6,1,6] >;
C4○D20 in GAP, Magma, Sage, TeX
C_4\circ D_{20} % in TeX
G:=Group("C4oD20"); // GroupNames label
G:=SmallGroup(80,38);
// by ID
G=gap.SmallGroup(80,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,1604]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^10=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^9>;
// generators/relations
Export
Subgroup lattice of C4○D20 in TeX
Character table of C4○D20 in TeX