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G = Dic20  order 80 = 24·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic20, C8.D5, C5⋊1Q16, 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 >

C1
C2
C5
C4
10C4
10C4
C10
C8
5Q8
5Q8
C20
2Dic5
2Dic5
5Q16
Dic10
C40
Dic10
Dic20

Character table of Dic20

 class 124A4B4C5A5B8A8B10A10B20A20B20C20D40A40B40C40D40E40F40G40H
 size 1122020222222222222222222
ρ111111111111111111111111    trivial
ρ2111-1-1111111111111111111    linear of order 2
ρ3111-1111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ522-200220022-2-2-2-200000000    orthogonal lifted from D4
ρ622200-1+√5/2-1-√5/222-1-√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
ρ722200-1-√5/2-1+√5/2-2-2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21+√5/21+√5/21-√5/21-√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ822200-1-√5/2-1+√5/222-1+√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
ρ922200-1+√5/2-1-√5/2-2-2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21-√5/21-√5/21+√5/21+√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ1022-200-1-√5/2-1+√5/200-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ1122-200-1+√5/2-1-√5/200-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ1222-200-1+√5/2-1-√5/200-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ1322-200-1-√5/2-1+√5/200-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ142-200022-√2√2-2-20000-√2-√2√2√2√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ152-200022√2-√2-2-20000√2√2-√2-√2-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ162-2000-1-√5/2-1+√5/2√2-√21-√5/21+√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52-ζ83ζ52+ζ8ζ53-ζ83ζ53+ζ8ζ52ζ83ζ53-ζ8ζ52-ζ87ζ5+ζ85ζ54ζ83ζ54-ζ8ζ5ζ83ζ52-ζ8ζ53ζ87ζ5-ζ85ζ54-ζ83ζ54+ζ8ζ5    symplectic faithful, Schur index 2
ρ172-2000-1-√5/2-1+√5/2-√2√21-√5/21+√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53-ζ83ζ52+ζ8ζ53-ζ83ζ54+ζ8ζ5ζ87ζ5-ζ85ζ54-ζ83ζ53+ζ8ζ52ζ83ζ54-ζ8ζ5-ζ87ζ5+ζ85ζ54    symplectic faithful, Schur index 2
ρ182-2000-1+√5/2-1-√5/2-√2√21+√5/21-√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5-ζ87ζ5+ζ85ζ54ζ83ζ54-ζ8ζ5-ζ83ζ54+ζ8ζ5-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53ζ87ζ5-ζ85ζ54ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53    symplectic faithful, Schur index 2
ρ192-2000-1+√5/2-1-√5/2√2-√21+√5/21-√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5ζ87ζ5-ζ85ζ54-ζ83ζ54+ζ8ζ5ζ83ζ54-ζ8ζ5ζ83ζ53-ζ8ζ52ζ83ζ52-ζ8ζ53-ζ87ζ5+ζ85ζ54-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53    symplectic faithful, Schur index 2
ρ202-2000-1+√5/2-1-√5/2-√2√21+√5/21-√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5ζ83ζ54-ζ8ζ5-ζ87ζ5+ζ85ζ54ζ87ζ5-ζ85ζ54-ζ83ζ52+ζ8ζ53-ζ83ζ53+ζ8ζ52-ζ83ζ54+ζ8ζ5ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52    symplectic faithful, Schur index 2
ρ212-2000-1-√5/2-1+√5/2-√2√21-√5/21+√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52-ζ83ζ53+ζ8ζ52ζ87ζ5-ζ85ζ54-ζ83ζ54+ζ8ζ5-ζ83ζ52+ζ8ζ53-ζ87ζ5+ζ85ζ54ζ83ζ54-ζ8ζ5    symplectic faithful, Schur index 2
ρ222-2000-1-√5/2-1+√5/2√2-√21-√5/21+√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52-ζ83ζ53+ζ8ζ52-ζ83ζ52+ζ8ζ53ζ83ζ52-ζ8ζ53ζ83ζ54-ζ8ζ5-ζ87ζ5+ζ85ζ54ζ83ζ53-ζ8ζ52-ζ83ζ54+ζ8ζ5ζ87ζ5-ζ85ζ54    symplectic faithful, Schur index 2
ρ232-2000-1+√5/2-1-√5/2√2-√21+√5/21-√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5-ζ83ζ54+ζ8ζ5ζ87ζ5-ζ85ζ54-ζ87ζ5+ζ85ζ54ζ83ζ52-ζ8ζ53ζ83ζ53-ζ8ζ52ζ83ζ54-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  D40⋊7C2  C8.D10  D8⋊3D5  SD16⋊D5  D5×Q16  C3⋊Dic20  Dic60  Dic100  C52⋊3Q16  C40.D5
Dic20 is a maximal quotient of
 C20.44D4  C40⋊5C4  C3⋊Dic20  Dic60  Dic100  C52⋊3Q16  C40.D5

Matrix representation of Dic20 ►in GL2(𝔽41) generated by

2320
185
,
1414
2427
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
 

Export

Subgroup lattice of Dic20 in TeX
Character table of Dic20 in TeX

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