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## G = Dic20order 80 = 24·5

### Dicyclic group

Aliases: Dic20, C8.D5, C51Q16, C40.1C2, C2.5D20, C10.3D4, C4.10D10, C20.10C22, Dic10.1C2, SmallGroup(80,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic20
 Chief series C1 — C5 — C10 — C20 — Dic10 — Dic20
 Lower central C5 — C10 — C20 — Dic20
 Upper central C1 — C2 — C4 — C8

Generators and relations for Dic20
G = < a,b | a40=1, b2=a20, bab-1=a-1 >

Character table of Dic20

 class 1 2 4A 4B 4C 5A 5B 8A 8B 10A 10B 20A 20B 20C 20D 40A 40B 40C 40D 40E 40F 40G 40H size 1 1 2 20 20 2 2 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 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 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 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 linear of order 2 ρ5 2 2 -2 0 0 2 2 0 0 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 0 0 -1+√5/2 -1-√5/2 2 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 ρ7 2 2 2 0 0 -1-√5/2 -1+√5/2 -2 -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 ρ8 2 2 2 0 0 -1-√5/2 -1+√5/2 2 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 ρ9 2 2 2 0 0 -1+√5/2 -1-√5/2 -2 -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 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ11 2 2 -2 0 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ12 2 2 -2 0 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ13 2 2 -2 0 0 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ14 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 -√2 -√2 √2 √2 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 √2 √2 -√2 -√2 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 0 0 0 -1-√5/2 -1+√5/2 √2 -√2 1-√5/2 1+√5/2 ζ82ζ53-ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ86ζ54+ζ86ζ5 -ζ82ζ53+ζ82ζ52 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 ζ83ζ53-ζ8ζ52 -ζ87ζ5+ζ85ζ54 ζ83ζ54-ζ8ζ5 ζ83ζ52-ζ8ζ53 ζ87ζ5-ζ85ζ54 -ζ83ζ54+ζ8ζ5 symplectic faithful, Schur index 2 ρ17 2 -2 0 0 0 -1-√5/2 -1+√5/2 -√2 √2 1-√5/2 1+√5/2 -ζ82ζ53+ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ86ζ54-ζ86ζ5 ζ82ζ53-ζ82ζ52 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 -ζ83ζ52+ζ8ζ53 -ζ83ζ54+ζ8ζ5 ζ87ζ5-ζ85ζ54 -ζ83ζ53+ζ8ζ52 ζ83ζ54-ζ8ζ5 -ζ87ζ5+ζ85ζ54 symplectic faithful, Schur index 2 ρ18 2 -2 0 0 0 -1+√5/2 -1-√5/2 -√2 √2 1+√5/2 1-√5/2 -ζ86ζ54+ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ82ζ53+ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ87ζ5+ζ85ζ54 ζ83ζ54-ζ8ζ5 -ζ83ζ54+ζ8ζ5 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 ζ87ζ5-ζ85ζ54 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 symplectic faithful, Schur index 2 ρ19 2 -2 0 0 0 -1+√5/2 -1-√5/2 √2 -√2 1+√5/2 1-√5/2 -ζ86ζ54+ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ82ζ53+ζ82ζ52 ζ86ζ54-ζ86ζ5 ζ87ζ5-ζ85ζ54 -ζ83ζ54+ζ8ζ5 ζ83ζ54-ζ8ζ5 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 -ζ87ζ5+ζ85ζ54 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 symplectic faithful, Schur index 2 ρ20 2 -2 0 0 0 -1+√5/2 -1-√5/2 -√2 √2 1+√5/2 1-√5/2 ζ86ζ54-ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ82ζ53-ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ83ζ54-ζ8ζ5 -ζ87ζ5+ζ85ζ54 ζ87ζ5-ζ85ζ54 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 -ζ83ζ54+ζ8ζ5 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 symplectic faithful, Schur index 2 ρ21 2 -2 0 0 0 -1-√5/2 -1+√5/2 -√2 √2 1-√5/2 1+√5/2 ζ82ζ53-ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ86ζ54+ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 -ζ83ζ53+ζ8ζ52 ζ87ζ5-ζ85ζ54 -ζ83ζ54+ζ8ζ5 -ζ83ζ52+ζ8ζ53 -ζ87ζ5+ζ85ζ54 ζ83ζ54-ζ8ζ5 symplectic faithful, Schur index 2 ρ22 2 -2 0 0 0 -1-√5/2 -1+√5/2 √2 -√2 1-√5/2 1+√5/2 -ζ82ζ53+ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ86ζ54-ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 ζ83ζ52-ζ8ζ53 ζ83ζ54-ζ8ζ5 -ζ87ζ5+ζ85ζ54 ζ83ζ53-ζ8ζ52 -ζ83ζ54+ζ8ζ5 ζ87ζ5-ζ85ζ54 symplectic faithful, Schur index 2 ρ23 2 -2 0 0 0 -1+√5/2 -1-√5/2 √2 -√2 1+√5/2 1-√5/2 ζ86ζ54-ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ82ζ53-ζ82ζ52 -ζ86ζ54+ζ86ζ5 -ζ83ζ54+ζ8ζ5 ζ87ζ5-ζ85ζ54 -ζ87ζ5+ζ85ζ54 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 ζ83ζ54-ζ8ζ5 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 symplectic faithful, Schur index 2

Smallest permutation representation of Dic20
Regular action on 80 points
Generators in S80
(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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 56 21 76)(2 55 22 75)(3 54 23 74)(4 53 24 73)(5 52 25 72)(6 51 26 71)(7 50 27 70)(8 49 28 69)(9 48 29 68)(10 47 30 67)(11 46 31 66)(12 45 32 65)(13 44 33 64)(14 43 34 63)(15 42 35 62)(16 41 36 61)(17 80 37 60)(18 79 38 59)(19 78 39 58)(20 77 40 57)

G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,56,21,76)(2,55,22,75)(3,54,23,74)(4,53,24,73)(5,52,25,72)(6,51,26,71)(7,50,27,70)(8,49,28,69)(9,48,29,68)(10,47,30,67)(11,46,31,66)(12,45,32,65)(13,44,33,64)(14,43,34,63)(15,42,35,62)(16,41,36,61)(17,80,37,60)(18,79,38,59)(19,78,39,58)(20,77,40,57)>;

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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,56,21,76)(2,55,22,75)(3,54,23,74)(4,53,24,73)(5,52,25,72)(6,51,26,71)(7,50,27,70)(8,49,28,69)(9,48,29,68)(10,47,30,67)(11,46,31,66)(12,45,32,65)(13,44,33,64)(14,43,34,63)(15,42,35,62)(16,41,36,61)(17,80,37,60)(18,79,38,59)(19,78,39,58)(20,77,40,57) );

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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,56,21,76),(2,55,22,75),(3,54,23,74),(4,53,24,73),(5,52,25,72),(6,51,26,71),(7,50,27,70),(8,49,28,69),(9,48,29,68),(10,47,30,67),(11,46,31,66),(12,45,32,65),(13,44,33,64),(14,43,34,63),(15,42,35,62),(16,41,36,61),(17,80,37,60),(18,79,38,59),(19,78,39,58),(20,77,40,57)]])

Dic20 is a maximal subgroup of
C16⋊D5  Dic40  D8.D5  C5⋊Q32  D407C2  C8.D10  D83D5  SD16⋊D5  D5×Q16  C3⋊Dic20  Dic60  Dic100  C523Q16  C40.D5
Dic20 is a maximal quotient of
C20.44D4  C405C4  C3⋊Dic20  Dic60  Dic100  C523Q16  C40.D5

Matrix representation of Dic20 in GL2(𝔽41) generated by

 23 20 18 5
,
 14 14 24 27
G:=sub<GL(2,GF(41))| [23,18,20,5],[14,24,14,27] >;

Dic20 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}
% in TeX

G:=Group("Dic20");
// GroupNames label

G:=SmallGroup(80,8);
// by ID

G=gap.SmallGroup(80,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,61,66,182,42,1604]);
// Polycyclic

G:=Group<a,b|a^40=1,b^2=a^20,b*a*b^-1=a^-1>;
// generators/relations

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