D4.D10 - GroupNames

G = D₄.D₁₀ order 160 = 2⁵·5

1^st non-split extension by D₄ of D₁₀ acting via D₁₀/C₁₀=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄.₆D₁₀, C₂₀.₁₅D₄, D₂₀:₆C₂², C₂₀.₁₂C₂³, Dic₁₀:₅C₂², D₄:D₅:₅C₂, (C₂xD₄):₂D₅, C₄oD₂₀:₃C₂, (D₄xC₁₀):₂C₂, C₅:₄(C₈:C₂²), D₄.D₅:₅C₂, C₅:₂C₈:₃C₂², C₁₀.₄₅(C₂xD₄), (C₂xC₄).₁₇D₁₀, (C₂xC₁₀).₃₉D₄, C₄.Dic₅:₆C₂, C₄.₁₆(C₅:D₄), (C₅xD₄).₆C₂², C₄.₁₂(C₂²xD₅), (C₂xC₂₀).₃₀C₂², C₂².₁₀(C₅:D₄), C₂.₉(C₂xC₅:D₄), SmallGroup(160,153)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₄.D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — D₂₀ — C₄oD₂₀ — D₄.D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — D₄.D₁₀

Upper central

C₁ — C₂ — C₂xC₄ — C₂xD₄

Generators and relations for D₄.D₁₀
G = < a,b,c,d | a⁴=b²=1, c¹⁰=d²=a², bab=dad^-1=a^-1, ac=ca, cbc^-1=a²b, dbd^-1=a^-1b, dcd^-1=c⁹ >

Subgroups: 208 in 68 conjugacy classes, 29 normal (19 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, M₄(2), D₈, SD₁₆, C₂xD₄, C₄oD₄, Dic₅, C₂₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₈:C₂², C₅:₂C₈, Dic₁₀, C₄xD₅, D₂₀, C₅:D₄, C₂xC₂₀, C₅xD₄, C₅xD₄, C₂²xC₁₀, C₄.Dic₅, D₄:D₅, D₄.D₅, C₄oD₂₀, D₄xC₁₀, D₄.D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, D₁₀, C₈:C₂², C₅:D₄, C₂²xD₅, C₂xC₅:D₄, D₄.D₁₀

Smallest permutation representation of D₄.D₁₀
►On 40 points

Generators in S₄₀

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

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

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

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

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

D₄.D₁₀ is a maximal subgroup of
D₂₀.₂D₄ D₂₀.₃D₄ D₂₀.₁₄D₄ D₂₀:₅D₄ C₄₀.₂₃D₄ C₄₀.₄₄D₄ D₂₀:₁₈D₄ D₂₀.₃₈D₄ D₈:₁₃D₁₀ D₂₀.₂₉D₄ D₅xC₈:C₂² SD₁₆:D₁₀ C₂₀.C₂⁴ D₂₀.₃₂C₂³ D₂₀.₃₃C₂³ D₂₀:₂₁D₆ D₆₀:₃₆C₂² D₂₀:₁₀D₆ D₁₂.₉D₁₀ D₄.D₃₀
D₄.D₁₀ is a maximal quotient of
C₂₀.₄₇(C₄:C₄) C₄oD₂₀:₉C₄ C₄:C₄.₂₂₈D₁₀ C₄:C₄.₂₃₀D₁₀ D₄.₃Dic₁₀ C₄².₄₈D₁₀ D₄.₁D₂₀ C₄².₅₁D₁₀ (C₂xD₄).D₁₀ D₂₀:₁₇D₄ C₄:D₄:D₅ C₄.(D₄xD₅) C₄².₇₂D₁₀ C₂₀:₂D₈ C₄².₇₄D₁₀ Dic₁₀:₉D₄ C₄².₇₆D₁₀ D₂₀:₅Q₈ C₄².₈₂D₁₀ Dic₁₀:₅Q₈ (D₄xC₁₀):₁₈C₄ (C₂xC₁₀):₈D₈ (C₅xD₄).₃₁D₄ D₂₀:₂₁D₆ D₆₀:₃₆C₂² D₂₀:₁₀D₆ D₁₂.₉D₁₀ D₄.D₃₀

31 conjugacy classes

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

31 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₅:D₄

C₈:C₂²

D₄.D₁₀

kernel

D₄.D₁₀

C₄.Dic₅

D₄:D₅

D₄.D₅

C₄oD₂₀

D₄xC₁₀

C₂₀

C₂xC₁₀

C₂xD₄

C₂xC₄

D₄

C₄

C₂²

C₅

C₁

# reps

Matrix representation of D₄.D₁₀ ►in GL₆(F₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	38	0	0
0	0	28	1	0	0
0	0	0	0	1	3
0	0	0	0	13	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	38	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	28	1

0	6	0	0	0	0
34	7	0	0	0	0
0	0	40	38	0	0
0	0	28	1	0	0
0	0	0	0	40	38
0	0	0	0	28	1

9	16	0	0	0	0
36	32	0	0	0	0
0	0	0	0	40	38
0	0	0	0	28	1
0	0	40	38	0	0
0	0	28	1	0	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,1,13,0,0,0,0,3,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,0,1],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,0,0,0,0,40,28,0,0,0,0,38,1],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,0,0,40,28,0,0,0,0,38,1,0,0,40,28,0,0,0,0,38,1,0,0] >;

D₄.D₁₀ in GAP, Magma, Sage, TeX

D_4.D_{10}

% in TeX

G:=Group("D4.D10");

// GroupNames label

G:=SmallGroup(160,153);

// by ID

G=gap.SmallGroup(160,153);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,579,159,69,4613]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^9>;

// generators/relations

G = D4.D10 order 160 = 25·5

1st non-split extension by D4 of D10 acting via D10/C10=C2

G = D₄.D₁₀ order 160 = 2⁵·5

1^st non-split extension by D₄ of D₁₀ acting via D₁₀/C₁₀=C₂