D20:1D4 - GroupNames

G = D₂₀⋊₁D₄ order 320 = 2⁶·5

1^st semidirect product of D₂₀ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₀⋊₁D₄, Dic₁₀⋊₁D₄, C₂³.₇D₂₀, M₄(2)⋊₂D₁₀, C₄.₇₉(D₄×D₅), C₂₀⋊D₄⋊₁C₂, C₈⋊D₁₀⋊₅C₂, C₅⋊₁(D₄⋊₄D₄), C₂₀.₉₂(C₂×D₄), C₄.D₄⋊₂D₅, D₂₀⋊₇C₄⋊₂C₂, D₄⋊₆D₁₀⋊₂C₂, (C₂×D₄).₁₄D₁₀, (C₂×C₂₀).₄C₂³, C₁₀.₁₆C₂²≀C₂, (C₂×D₂₀)⋊₁₁C₂², (C₄×Dic₅)⋊₂C₂², C₄○D₂₀.₂C₂², (C₂²×C₁₀).₂₀D₄, C₂².₁₁(C₂×D₂₀), (D₄×C₁₀).₁₄C₂², (C₅×M₄(2))⋊₉C₂², C₂.₁₉(C₂²⋊D₂₀), (C₅×C₄.D₄)⋊₄C₂, (C₂×C₁₀).₂₁(C₂×D₄), (C₂×C₄).₄(C₂²×D₅), SmallGroup(320,374)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — D₂₀⋊₁D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄○D₂₀ — D₄⋊₆D₁₀ — D₂₀⋊₁D₄

Lower central

C₅ — C₁₀ — C₂×C₂₀ — D₂₀⋊₁D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₄.D₄

Generators and relations for D₂₀⋊₁D₄
G = < a,b,c,d | a²⁰=b²=c⁴=d²=1, bab=dad=a^-1, cac^-1=a⁹, cbc^-1=a¹³b, dbd=a³b, dcd=c^-1 >

Subgroups: 942 in 168 conjugacy classes, 39 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₄₀, Dic₁₀, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, D₄⋊₄D₄, C₄₀⋊C₂, D₄₀, C₄×Dic₅, C₅×M₄(2), C₂×D₂₀, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, C₂×C₅⋊D₄, D₄×C₁₀, D₂₀⋊₇C₄, C₅×C₄.D₄, C₈⋊D₁₀, C₂₀⋊D₄, D₄⋊₆D₁₀, D₂₀⋊₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, D₁₀, C₂²≀C₂, D₂₀, C₂²×D₅, D₄⋊₄D₄, C₂×D₂₀, D₄×D₅, C₂²⋊D₂₀, D₂₀⋊₁D₄

Smallest permutation representation of D₂₀⋊₁D₄
►On 40 points

Generators in S₄₀

(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 40)(18 39)(19 38)(20 37)

(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)

(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)

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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)>;

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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35) );

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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,40),(18,39),(19,38),(20,37)], [(1,6,11,16),(2,15,12,5),(3,4,13,14),(7,20,17,10),(8,9,18,19),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35)]])

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₂₀

D₄⋊₄D₄

D₄×D₅

D₂₀⋊₁D₄

kernel

D₂₀⋊₁D₄

D₂₀⋊₇C₄

C₅×C₄.D₄

C₈⋊D₁₀

C₂₀⋊D₄

D₄⋊₆D₁₀

Dic₁₀

D₂₀

C₂²×C₁₀

C₄.D₄

M₄(2)

C₂×D₄

C₂³

C₅

C₄

C₁

# reps

Matrix representation of D₂₀⋊₁D₄ ►in GL₆(𝔽₄₁)

6	40	0	0	0	0
36	1	0	0	0	0
0	0	0	40	0	0
0	0	1	0	0	0
0	0	28	28	40	5
0	0	0	3	16	1

2	11	0	0	0	0
37	39	0	0	0	0
0	0	0	0	1	0
0	0	28	28	40	5
0	0	1	0	0	0
0	0	32	32	0	13

35	34	0	0	0	0
5	6	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	0	40	0
0	0	0	38	0	40

35	34	0	0	0	0
5	6	0	0	0	0
0	0	40	0	0	0
0	0	0	1	0	0
0	0	28	28	40	5
0	0	3	0	0	1

G:=sub<GL(6,GF(41))| [6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,1,28,0,0,0,40,0,28,3,0,0,0,0,40,16,0,0,0,0,5,1],[2,37,0,0,0,0,11,39,0,0,0,0,0,0,0,28,1,32,0,0,0,28,0,32,0,0,1,40,0,0,0,0,0,5,0,13],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,38,0,0,0,0,40,0,0,0,0,0,0,40],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,40,0,28,3,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,5,1] >;

D₂₀⋊₁D₄ in GAP, Magma, Sage, TeX

D_{20}\rtimes_1D_4

% in TeX

G:=Group("D20:1D4");

// GroupNames label

G:=SmallGroup(320,374);

// by ID

G=gap.SmallGroup(320,374);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,1123,570,136,1684,438,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^3*b,d*c*d=c^-1>;

// generators/relations

G = D20⋊1D4 order 320 = 26·5

1st semidirect product of D20 and D4 acting via D4/C2=C22

G = D₂₀⋊₁D₄ order 320 = 2⁶·5

1^st semidirect product of D₂₀ and D₄ acting via D₄/C₂=C₂²