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G = Dic20order 80 = 24·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C20 — Dic20
C1C5C10C20Dic10 — Dic20
C5C10C20 — Dic20
C1C2C4C8

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

10C4
10C4
5Q8
5Q8
2Dic5
2Dic5
5Q16

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+5/2ζ4ζ534ζ524ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    orthogonal lifted from D20
ρ1122-200-1+5/2-1-5/200-1-5/2-1+5/21-5/21+5/21+5/21-5/243ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    orthogonal lifted from D20
ρ1222-200-1+5/2-1-5/200-1-5/2-1+5/21-5/21+5/21+5/21-5/2ζ43ζ5443ζ543ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ1322-200-1-5/2-1+5/200-1+5/2-1-5/21+5/21-5/21-5/21+5/24ζ534ζ52ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    orthogonal lifted from D20
ρ142-200022-22-2-20000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ152-2000222-2-2-2000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ162-2000-1-5/2-1+5/22-21-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ5283ζ528ζ5383ζ538ζ52ζ83ζ538ζ5287ζ585ζ54ζ83ζ548ζ5ζ83ζ528ζ53ζ87ζ585ζ5483ζ548ζ5    symplectic faithful, Schur index 2
ρ172-2000-1-5/2-1+5/2-221-5/21+5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ83ζ538ζ52ζ83ζ528ζ5383ζ528ζ5383ζ548ζ5ζ87ζ585ζ5483ζ538ζ52ζ83ζ548ζ587ζ585ζ54    symplectic faithful, Schur index 2
ρ182-2000-1+5/2-1-5/2-221+5/21-5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ587ζ585ζ54ζ83ζ548ζ583ζ548ζ583ζ538ζ5283ζ528ζ53ζ87ζ585ζ54ζ83ζ538ζ52ζ83ζ528ζ53    symplectic faithful, Schur index 2
ρ192-2000-1+5/2-1-5/22-21+5/21-5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ5ζ87ζ585ζ5483ζ548ζ5ζ83ζ548ζ5ζ83ζ538ζ52ζ83ζ528ζ5387ζ585ζ5483ζ538ζ5283ζ528ζ53    symplectic faithful, Schur index 2
ρ202-2000-1+5/2-1-5/2-221+5/21-5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ5ζ83ζ548ζ587ζ585ζ54ζ87ζ585ζ5483ζ528ζ5383ζ538ζ5283ζ548ζ5ζ83ζ528ζ53ζ83ζ538ζ52    symplectic faithful, Schur index 2
ρ212-2000-1-5/2-1+5/2-221-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ52ζ83ζ528ζ53ζ83ζ538ζ5283ζ538ζ52ζ87ζ585ζ5483ζ548ζ583ζ528ζ5387ζ585ζ54ζ83ζ548ζ5    symplectic faithful, Schur index 2
ρ222-2000-1-5/2-1+5/22-21-5/21+5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ5283ζ538ζ5283ζ528ζ53ζ83ζ528ζ53ζ83ζ548ζ587ζ585ζ54ζ83ζ538ζ5283ζ548ζ5ζ87ζ585ζ54    symplectic faithful, Schur index 2
ρ232-2000-1+5/2-1-5/22-21+5/21-5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ583ζ548ζ5ζ87ζ585ζ5487ζ585ζ54ζ83ζ528ζ53ζ83ζ538ζ52ζ83ζ548ζ583ζ528ζ5383ζ538ζ52    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 49 21 69)(2 48 22 68)(3 47 23 67)(4 46 24 66)(5 45 25 65)(6 44 26 64)(7 43 27 63)(8 42 28 62)(9 41 29 61)(10 80 30 60)(11 79 31 59)(12 78 32 58)(13 77 33 57)(14 76 34 56)(15 75 35 55)(16 74 36 54)(17 73 37 53)(18 72 38 52)(19 71 39 51)(20 70 40 50)

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

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

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

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

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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