Dic10:D5 - GroupNames

G = Dic₁₀:D₅ order 400 = 2⁴·5²

4^th semidirect product of Dic₁₀ and D₅ acting via D₅/C₅=C₂

Aliases: Dic₁₀:₄D₅, C₂₀.₂₃D₁₀, Dic₅.₂D₁₀, C₄.₁₂D₅², C₅:D₅:₂Q₈, C₅:₁(Q₈xD₅), C₅²:₃(C₂xQ₈), C₅²:₂Q₈:₃C₂, (C₅xDic₁₀):₆C₂, (C₅xC₁₀).₄C₂³, C₁₀.₄(C₂²xD₅), (C₅xC₂₀).₁₉C₂², Dic₅:₂D₅.₁C₂, (C₅xDic₅).₃C₂², C₅²:₆C₄.₁₀C₂², C₂.₇(C₂xD₅²), (C₄xC₅:D₅).₁C₂, (C₂xC₅:D₅).₁₃C₂², SmallGroup(400,166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅xC₁₀ — Dic₁₀:D₅

Chief series

C₁ — C₅ — C₅² — C₅xC₁₀ — C₅xDic₅ — Dic₅:₂D₅ — Dic₁₀:D₅

Lower central

C₅² — C₅xC₁₀ — Dic₁₀:D₅

Upper central

C₁ — C₂ — C₄

Generators and relations for Dic₁₀:D₅
G = < a,b,c,d | a²⁰=c⁵=d²=1, b²=a¹⁰, bab^-1=a^-1, ac=ca, dad=a⁹, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 556 in 92 conjugacy classes, 34 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₅, C₂xC₄, Q₈, D₅, C₁₀, C₁₀, C₂xQ₈, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, C₅², Dic₁₀, Dic₁₀, C₄xD₅, C₅xQ₈, C₅:D₅, C₅xC₁₀, Q₈xD₅, C₅xDic₅, C₅²:₆C₄, C₅xC₂₀, C₂xC₅:D₅, Dic₅:₂D₅, C₅²:₂Q₈, C₅xDic₁₀, C₄xC₅:D₅, Dic₁₀:D₅
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂xQ₈, D₁₀, C₂²xD₅, Q₈xD₅, D₅², C₂xD₅², Dic₁₀:D₅

Smallest permutation representation of Dic₁₀:D₅
►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 25 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)

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

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

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

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

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

46 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	5A	5B	5C	5D	5E	5F	5G	5H	10A	10B	10C	10D	10E	10F	10G	10H	20A	···	20L	20M	···	20T
order	1	2	2	2	4	4	4	4	4	4	5	5	5	5	5	5	5	5	10	10	10	10	10	10	10	10	20	···	20	20	···	20
size	1	1	25	25	2	10	10	10	10	50	2	2	2	2	4	4	4	4	2	2	2	2	4	4	4	4	4	···	4	20	···	20

46 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+	+	+	-	+	+	+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	Q₈	D₅	D₁₀	D₁₀	Q₈xD₅	D₅²	C₂xD₅²	Dic₁₀:D₅
kernel	Dic₁₀:D₅	Dic₅:₂D₅	C₅²:₂Q₈	C₅xDic₁₀	C₄xC₅:D₅	C₅:D₅	Dic₁₀	Dic₅	C₂₀	C₅	C₄	C₂	C₁
# reps	1	2	2	2	1	2	4	8	4	4	4	4	8

Matrix representation of Dic₁₀:D₅ ►in GL₆(F₄₁)

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

40	0	0	0	0	0
7	1	0	0	0	0
0	0	15	15	0	0
0	0	15	26	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	40
0	0	0	0	1	6

1	0	0	0	0	0
34	40	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	6
0	0	0	0	0	40

G:=sub<GL(6,GF(41))| [34,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,6,40] >;

Dic₁₀:D₅ in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes D_5

% in TeX

G:=Group("Dic10:D5");

// GroupNames label

G:=SmallGroup(400,166);

// by ID

G=gap.SmallGroup(400,166);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,55,218,116,50,970,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^20=c^5=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = Dic10:D5 order 400 = 24·52

4th semidirect product of Dic10 and D5 acting via D5/C5=C2

G = Dic₁₀:D₅ order 400 = 2⁴·5²

4^th semidirect product of Dic₁₀ and D₅ acting via D₅/C₅=C₂