Dic5:2D5 - GroupNames

(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

(1 12 6 17)(2 11 7 16)(3 20 8 15)(4 19 9 14)(5 18 10 13)

(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)

(1 9)(2 8)(3 7)(4 6)(11 15)(12 14)(16 20)(17 19)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12,6,17)(2,11,7,16)(3,20,8,15)(4,19,9,14)(5,18,10,13), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,9)(2,8)(3,7)(4,6)(11,15)(12,14)(16,20)(17,19)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12,6,17)(2,11,7,16)(3,20,8,15)(4,19,9,14)(5,18,10,13), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,9)(2,8)(3,7)(4,6)(11,15)(12,14)(16,20)(17,19) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12,6,17),(2,11,7,16),(3,20,8,15),(4,19,9,14),(5,18,10,13)], [(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14)], [(1,9),(2,8),(3,7),(4,6),(11,15),(12,14),(16,20),(17,19)]])

G:=TransitiveGroup(20,58);

Dic₅:₂D₅ is a maximal subgroup of
Dic₅.₄F₅ Dic₅.F₅ C₅²:₃C₄² Dic₅:F₅ D₅²:C₄ C₂.D₅wrC₂ Dic₁₀:D₅ Dic₁₀:₅D₅ C₄xD₅² Dic₅.D₁₀ D₁₀:D₁₀
Dic₅:₂D₅ is a maximal quotient of
C₂₀.₂₉D₁₀ C₂₀.₃₁D₁₀ Dic₅² C₁₀.D₂₀ C₁₀.Dic₁₀

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	2	2	2	4	4
type	+	+	+		+	+		+	+
image	C₁	C₂	C₂	C₄	D₅	D₁₀	C₄xD₅	D₅²	Dic₅:₂D₅
kernel	Dic₅:₂D₅	C₅xDic₅	C₂xC₅:D₅	C₅:D₅	Dic₅	C₁₀	C₅	C₂	C₁
# reps	1	2	1	4	4	4	8	4	4

Matrix representation of Dic₅:₂D₅ ►in GL₄(F₄₁) generated by

35	1	0	0
40	0	0	0
0	0	1	0
0	0	0	1

32	0	0	0
28	9	0	0
0	0	1	0
0	0	0	1

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

1	0	0	0
6	40	0	0
0	0	6	40
0	0	35	35

G:=sub<GL(4,GF(41))| [35,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[32,28,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[1,6,0,0,0,40,0,0,0,0,6,35,0,0,40,35] >;

Dic₅:₂D₅ in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_2D_5

% in TeX

G:=Group("Dic5:2D5");

// GroupNames label

G:=SmallGroup(200,23);

// by ID

G=gap.SmallGroup(200,23);

# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,26,328,4004]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of Dic₅:₂D₅ in TeX

G = Dic5:2D5 order 200 = 23·52

The semidirect product of Dic5 and D5 acting through Inn(Dic5)

G = Dic₅:₂D₅ order 200 = 2³·5²

The semidirect product of Dic₅ and D₅ acting through Inn(Dic₅)