metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊1F5, D20⋊1C4, D5.2D8, D10.18D4, Dic5.2D4, D5.2SD16, C4⋊F5⋊1C2, D5⋊C8⋊1C2, C5⋊(D4⋊C4), (C5×D4)⋊1C4, C4.1(C2×F5), C20.1(C2×C4), (D4×D5).2C2, (C4×D5).7C22, C2.6(C22⋊F5), C10.5(C22⋊C4), SmallGroup(160,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a17b >
Character table of D20⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20 | |
| size | 1 | 1 | 4 | 5 | 5 | 20 | 2 | 10 | 20 | 20 | 4 | 10 | 10 | 10 | 10 | 4 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | i | -i | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | i | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | -√-2 | √-2 | √-2 | -2 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ14 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | √-2 | -√-2 | -√-2 | -2 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | orthogonal lifted from C22⋊F5 |
| ρ18 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | orthogonal lifted from C22⋊F5 |
| ρ19 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 40 points
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)
(1 37)(2 24 10 40)(3 31 19 23)(4 38 8 26)(5 25 17 29)(6 32)(7 39 15 35)(9 33 13 21)(11 27)(12 34 20 30)(14 28 18 36)(16 22)
G:=sub<Sym(40)| (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35), (1,37)(2,24,10,40)(3,31,19,23)(4,38,8,26)(5,25,17,29)(6,32)(7,39,15,35)(9,33,13,21)(11,27)(12,34,20,30)(14,28,18,36)(16,22)>;
G:=Group( (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35), (1,37)(2,24,10,40)(3,31,19,23)(4,38,8,26)(5,25,17,29)(6,32)(7,39,15,35)(9,33,13,21)(11,27)(12,34,20,30)(14,28,18,36)(16,22) );
G=PermutationGroup([[(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,27),(22,26),(23,25),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35)], [(1,37),(2,24,10,40),(3,31,19,23),(4,38,8,26),(5,25,17,29),(6,32),(7,39,15,35),(9,33,13,21),(11,27),(12,34,20,30),(14,28,18,36),(16,22)]])
D20⋊C4 is a maximal subgroup of
D8×F5 D40⋊C4 SD16×F5 SD16⋊F5 (D4×C10)⋊C4 C4○D4⋊F5 C4○D20⋊C4 D60⋊C4 D12⋊F5 D20⋊Dic3
D20⋊C4 is a maximal quotient of
D10.1D8 D20⋊C8 Dic5.D8 D10.18D8 D10.D8 D5.D16 D8.F5 D40.C4 D40⋊1C4 D5.Q32 Q16.F5 Dic20.C4 D10.SD16 Dic5.23D8 Dic5.SD16 D60⋊C4 D12⋊F5 D20⋊Dic3
Matrix representation of D20⋊C4 ►in GL6(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 35 | 35 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 40 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 24 | 0 | 0 |
| 0 | 0 | 18 | 38 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,35,0,0,0,0,7,35],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[0,15,0,0,0,0,30,0,0,0,0,0,0,0,0,0,3,18,0,0,0,0,24,38,0,0,2,0,0,0,0,0,0,2,0,0] >;
D20⋊C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes C_4 % in TeX
G:=Group("D20:C4"); // GroupNames label
G:=SmallGroup(160,82);
// by ID
G=gap.SmallGroup(160,82);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,579,297,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;
// generators/relations
Export
Subgroup lattice of D20⋊C4 in TeX
Character table of D20⋊C4 in TeX