Dic5.D10 - GroupNames

G = Dic₅.D₁₀ order 400 = 2⁴·5²

3^rd non-split extension by Dic₅ of D₁₀ acting via D₁₀/D₅=C₂

Aliases: D₁₀.₃D₁₀, Dic₅.₃D₁₀, C₁₀².₇C₂², C₅:D₄:₃D₅, C₂².₁D₅², C₅:₄(C₄oD₂₀), C₅:D₂₀:₂C₂, (D₅xDic₅):₅C₂, (C₂xDic₅):₃D₅, (C₂xC₁₀).₃D₁₀, C₅²:₆(C₄oD₄), C₅²:₇D₄:₁C₂, C₅:₃(D₄:₂D₅), C₅²:₂Q₈:₄C₂, (C₁₀xDic₅):₆C₂, Dic₅:₂D₅:₂C₂, (C₅xC₁₀).₁₁C₂³, (D₅xC₁₀).₃C₂², C₁₀.₁₁(C₂²xD₅), C₅²:₆C₄.₃C₂², (C₅xDic₅).₂₁C₂², C₂.₁₂(C₂xD₅²), (C₅xC₅:D₄):₁C₂, (C₂xC₅:D₅).₂C₂², SmallGroup(400,173)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅xC₁₀ — Dic₅.D₁₀

Chief series

C₁ — C₅ — C₅² — C₅xC₁₀ — D₅xC₁₀ — D₅xDic₅ — Dic₅.D₁₀

Lower central

C₅² — C₅xC₁₀ — Dic₅.D₁₀

Upper central

C₁ — C₂ — C₂²

Generators and relations for Dic₅.D₁₀
G = < a,b,c,d | a¹⁰=c¹⁰=1, b²=d²=a⁵, bab^-1=cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=a⁵b, dcd^-1=c^-1 >

Subgroups: 596 in 96 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂xC₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₄oD₄, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₅², Dic₁₀, C₄xD₅, D₂₀, C₂xDic₅, C₂xDic₅, C₅:D₄, C₅:D₄, C₂xC₂₀, C₅xD₄, C₅xD₅, C₅:D₅, C₅xC₁₀, C₅xC₁₀, C₄oD₂₀, D₄:₂D₅, C₅xDic₅, C₅²:₆C₄, D₅xC₁₀, C₂xC₅:D₅, C₁₀², D₅xDic₅, Dic₅:₂D₅, C₅:D₂₀, C₅²:₂Q₈, C₁₀xDic₅, C₅xC₅:D₄, C₅²:₇D₄, Dic₅.D₁₀
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄oD₄, D₁₀, C₂²xD₅, C₄oD₂₀, D₄:₂D₅, D₅², C₂xD₅², Dic₅.D₁₀

Smallest permutation representation of Dic₅.D₁₀
►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 26 6 21)(2 25 7 30)(3 24 8 29)(4 23 9 28)(5 22 10 27)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)

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

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

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

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

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

52 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	5A	5B	5C	5D	5E	5F	5G	5H	10A	···	10H	10I	···	10V	10W	10X	20A	···	20H	20I	20J
order	1	2	2	2	2	4	4	4	4	4	5	5	5	5	5	5	5	5	10	···	10	10	···	10	10	10	20	···	20	20	20
size	1	1	2	10	50	5	5	10	10	50	2	2	2	2	4	4	4	4	2	···	2	4	···	4	20	20	10	···	10	20	20

52 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+	+		-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₅	D₅	C₄oD₄	D₁₀	D₁₀	D₁₀	C₄oD₂₀	D₄:₂D₅	D₅²	C₂xD₅²	Dic₅.D₁₀
kernel	Dic₅.D₁₀	D₅xDic₅	Dic₅:₂D₅	C₅:D₂₀	C₅²:₂Q₈	C₁₀xDic₅	C₅xC₅:D₄	C₅²:₇D₄	C₂xDic₅	C₅:D₄	C₅²	Dic₅	D₁₀	C₂xC₁₀	C₅	C₅	C₂²	C₂	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	6	2	4	8	2	4	4	8

Matrix representation of Dic₅.D₁₀ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	34	40
0	0	0	0	1	0

12	23	0	0	0	0
24	29	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	34	40

12	23	0	0	0	0
33	29	0	0	0	0
0	0	40	34	0	0
0	0	7	7	0	0
0	0	0	0	1	0
0	0	0	0	34	40

15	39	0	0	0	0
31	26	0	0	0	0
0	0	1	7	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	34	40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[12,24,0,0,0,0,23,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[12,33,0,0,0,0,23,29,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[15,31,0,0,0,0,39,26,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

Dic₅.D₁₀ in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_{10}

% in TeX

G:=Group("Dic5.D10");

// GroupNames label

G:=SmallGroup(400,173);

// by ID

G=gap.SmallGroup(400,173);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,970,11525]);

// Polycyclic

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

// generators/relations

G = Dic5.D10 order 400 = 24·52

3rd non-split extension by Dic5 of D10 acting via D10/D5=C2

G = Dic₅.D₁₀ order 400 = 2⁴·5²

3^rd non-split extension by Dic₅ of D₁₀ acting via D₁₀/D₅=C₂