D10.4D10 - GroupNames

G = D₁₀.₄D₁₀ order 400 = 2⁴·5²

4^th non-split extension by D₁₀ of D₁₀ acting via D₁₀/D₅=C₂

Aliases: D₁₀.₄D₁₀, Dic₅.₄D₁₀, C₁₀².₈C₂², C₅:D₄:₁D₅, C₂².₂D₅², (D₅xDic₅):₂C₂, (C₂xC₁₀).₄D₁₀, C₅²:₇(C₄oD₄), C₅²:₂D₄:₃C₂, C₅:₄(D₄:₂D₅), C₅²:₂Q₈:₅C₂, (C₅xC₁₀).₁₂C₂³, (D₅xC₁₀).₄C₂², C₁₀.₁₂(C₂²xD₅), (C₅xDic₅).₅C₂², C₅²:₆C₄.₁₂C₂², C₂.₁₃(C₂xD₅²), (C₅xC₅:D₄):₂C₂, (C₂xC₅²:₆C₄):₅C₂, SmallGroup(400,174)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅xC₁₀ — D₁₀.₄D₁₀

Chief series

C₁ — C₅ — C₅² — C₅xC₁₀ — D₅xC₁₀ — D₅xDic₅ — D₁₀.₄D₁₀

Lower central

C₅² — C₅xC₁₀ — D₁₀.₄D₁₀

Upper central

C₁ — C₂ — C₂²

Generators and relations for D₁₀.₄D₁₀
G = < a,b,c,d | a¹⁰=b²=1, c¹⁰=d²=a⁵, bab=cac^-1=dad^-1=a^-1, cbc^-1=a³b, dbd^-1=a⁸b, dcd^-1=c⁹ >

Subgroups: 508 in 96 conjugacy classes, 32 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂xC₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₄oD₄, Dic₅, Dic₅, C₂₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₅², Dic₁₀, C₄xD₅, C₂xDic₅, C₅:D₄, C₅:D₄, C₅xD₄, C₅xD₅, C₅xC₁₀, C₅xC₁₀, D₄:₂D₅, C₅xDic₅, C₅²:₆C₄, D₅xC₁₀, C₁₀², D₅xDic₅, C₅²:₂D₄, C₅²:₂Q₈, C₅xC₅:D₄, C₂xC₅²:₆C₄, D₁₀.₄D₁₀
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄oD₄, D₁₀, C₂²xD₅, D₄:₂D₅, D₅², C₂xD₅², D₁₀.₄D₁₀

Smallest permutation representation of D₁₀.₄D₁₀
►On 40 points

Generators in S₄₀

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

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

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

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

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

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

46 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	5A	5B	5C	5D	5E	5F	5G	5H	10A	10B	10C	10D	10E	···	10T	10U	10V	10W	10X	20A	20B	20C	20D
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	10	10	10	10	20	20	20	20
size	1	1	2	10	10	10	10	25	25	50	2	2	2	2	4	4	4	4	2	2	2	2	4	···	4	20	20	20	20	20	20	20	20

46 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+		+	+	+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₅	C₄oD₄	D₁₀	D₁₀	D₁₀	D₄:₂D₅	D₅²	C₂xD₅²	D₁₀.₄D₁₀
kernel	D₁₀.₄D₁₀	D₅xDic₅	C₅²:₂D₄	C₅²:₂Q₈	C₅xC₅:D₄	C₂xC₅²:₆C₄	C₅:D₄	C₅²	Dic₅	D₁₀	C₂xC₁₀	C₅	C₂²	C₂	C₁
# reps	1	2	1	1	2	1	4	2	4	4	4	4	4	4	8

Matrix representation of D₁₀.₄D₁₀ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	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	0	1

0	40	0	0	0	0
40	0	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	0	1

0	1	0	0	0	0
40	0	0	0	0	0
0	0	1	0	0	0
0	0	34	40	0	0
0	0	0	0	40	34
0	0	0	0	7	7

32	0	0	0	0	0
0	32	0	0	0	0
0	0	40	0	0	0
0	0	7	1	0	0
0	0	0	0	40	34
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,34,1] >;

D₁₀.₄D₁₀ in GAP, Magma, Sage, TeX

D_{10}._4D_{10}

% in TeX

G:=Group("D10.4D10");

// GroupNames label

G:=SmallGroup(400,174);

// by ID

G=gap.SmallGroup(400,174);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=1,c^10=d^2=a^5,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^8*b,d*c*d^-1=c^9>;

// generators/relations

G = D10.4D10 order 400 = 24·52

4th non-split extension by D10 of D10 acting via D10/D5=C2

G = D₁₀.₄D₁₀ order 400 = 2⁴·5²

4^th non-split extension by D₁₀ of D₁₀ acting via D₁₀/D₅=C₂