D20.3D4 - GroupNames

G = D₂₀.₃D₄ order 320 = 2⁶·5

3^rd non-split extension by D₂₀ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₀.₃D₄, Dic₁₀.₃D₄, M₄(2).₂D₁₀, C₈⋊D₁₀⋊₆C₂, C₄.D₄⋊₄D₅, C₄.₁₄₉(D₄×D₅), C₂₀.₉₄(C₂×D₄), C₅⋊₂C₈.₄₁D₄, (C₂×D₄).₁₆D₁₀, C₅⋊₁(D₄.₄D₄), (C₂×C₂₀).₆C₂³, D₂₀.₂C₄⋊₆C₂, D₄.D₁₀⋊₁C₂, C₂₀.₅₃D₄⋊₂C₂, C₂₀.₄₆D₄⋊₆C₂, C₄○D₂₀.₄C₂², C₁₀.₁₀(C₄⋊D₄), (D₄×C₁₀).₁₆C₂², (C₂×D₂₀).₄₃C₂², C₂.₁₃(D₁₀⋊D₄), C₄.Dic₅.₃C₂², C₂².₁₄(C₄○D₂₀), (C₅×M₄(2)).₁₁C₂², (C₂×D₄⋊D₅)⋊₁C₂, (C₅×C₄.D₄)⋊₂C₂, (C₂×C₄).₆(C₂²×D₅), (C₂×C₅⋊₂C₈).₂C₂², (C₂×C₁₀).₃₁(C₄○D₄), SmallGroup(320,376)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — D₂₀.₃D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄○D₂₀ — D₂₀.₂C₄ — D₂₀.₃D₄

Lower central

C₅ — C₁₀ — C₂×C₂₀ — D₂₀.₃D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₄.D₄

Generators and relations for D₂₀.₃D₄
G = < a,b,c,d | a²⁰=b²=1, c⁴=a¹⁰, d²=a¹⁵, bab=a^-1, cac^-1=a¹¹, ad=da, cbc^-1=a¹⁰b, dbd^-1=a¹⁵b, dcd^-1=a⁵c³ >

Subgroups: 510 in 108 conjugacy classes, 35 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄.D₄, C₈.C₄, C₈○D₄, C₂×D₈, C₈⋊C₂², C₅⋊₂C₈, C₅⋊₂C₈, C₄₀, Dic₁₀, C₄×D₅, D₂₀, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, D₄.₄D₄, C₈×D₅, C₈⋊D₅, C₄₀⋊C₂, D₄₀, C₂×C₅⋊₂C₈, C₄.Dic₅, D₄⋊D₅, D₄.D₅, C₅×M₄(2), C₂×D₂₀, C₄○D₂₀, D₄×C₁₀, C₂₀.₅₃D₄, C₂₀.₄₆D₄, C₅×C₄.D₄, D₂₀.₂C₄, C₈⋊D₁₀, C₂×D₄⋊D₅, D₄.D₁₀, D₂₀.₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₄○D₄, D₁₀, C₄⋊D₄, C₂²×D₅, D₄.₄D₄, C₄○D₂₀, D₄×D₅, D₁₀⋊D₄, D₂₀.₃D₄

Smallest permutation representation of D₂₀.₃D₄
►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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 60)(38 59)(39 58)(40 57)

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	5A	5B	8A	8B	8C	8D	8E	8F	8G	10A	10B	10C	10D	10E	10F	10G	10H	20A	20B	20C	20D	40A	···	40H
order	1	2	2	2	2	2	4	4	4	5	5	8	8	8	8	8	8	8	10	10	10	10	10	10	10	10	20	20	20	20	40	···	40
size	1	1	2	8	20	40	2	2	20	2	2	4	4	8	10	10	20	40	2	2	4	4	8	8	8	8	4	4	4	4	8	···	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

C₄○D₄

D₁₀

C₄○D₂₀

D₄.₄D₄

D₄×D₅

D₂₀.₃D₄

kernel

D₂₀.₃D₄

C₂₀.₅₃D₄

C₂₀.₄₆D₄

C₅×C₄.D₄

D₂₀.₂C₄

C₈⋊D₁₀

C₂×D₄⋊D₅

D₄.D₁₀

C₅⋊₂C₈

Dic₁₀

D₂₀

C₄.D₄

C₂×C₁₀

M₄(2)

C₂×D₄

C₂²

C₅

C₄

C₁

# reps

Matrix representation of D₂₀.₃D₄ ►in GL₆(𝔽₄₁)

34	1	0	0	0	0
40	0	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	4	0	40
0	0	37	0	1	0

1	34	0	0	0	0
0	40	0	0	0	0
0	0	29	29	17	17
0	0	15	26	24	17
0	0	35	40	15	12
0	0	13	35	26	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	36	39	0
0	0	36	4	0	39
0	0	0	0	37	5
0	0	1	0	5	37

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	26	24	17
0	0	29	29	17	17
0	0	13	35	26	12
0	0	35	40	15	12

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,29,15,35,13,0,0,29,26,40,35,0,0,17,24,15,26,0,0,17,17,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,36,0,1,0,0,36,4,0,0,0,0,39,0,37,5,0,0,0,39,5,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,29,13,35,0,0,26,29,35,40,0,0,24,17,26,15,0,0,17,17,12,12] >;

D₂₀.₃D₄ in GAP, Magma, Sage, TeX

D_{20}._3D_4

% in TeX

G:=Group("D20.3D4");

// GroupNames label

G:=SmallGroup(320,376);

// by ID

G=gap.SmallGroup(320,376);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,297,136,1684,851,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^3>;

// generators/relations

G = D20.3D4 order 320 = 26·5

3rd non-split extension by D20 of D4 acting via D4/C2=C22

G = D₂₀.₃D₄ order 320 = 2⁶·5

3^rd non-split extension by D₂₀ of D₄ acting via D₄/C₂=C₂²