Dic5.F5 - GroupNames

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

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

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

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

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

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

34 conjugacy classes

class	1	2A	2B	4A	4B	4C	5A	5B	5C	5D	5E	8A	8B	8C	8D	10A	10B	10C	10D	10E	20A	20B	20C	20D	20E	20F	40A	···	40H
order	1	2	2	4	4	4	5	5	5	5	5	8	8	8	8	10	10	10	10	10	20	20	20	20	20	20	40	···	40
size	1	1	50	5	5	10	2	2	4	8	8	10	10	50	50	2	2	4	8	8	10	10	10	10	20	20	10	···	10

34 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	8	8
type	+	+	+	+			+		+			+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₅	M₄(2)	D₁₀	C₄×D₅	C₈⋊D₅	F₅	C₂×F₅	C₄.F₅	D₅×F₅	Dic₅.F₅
kernel	Dic₅.F₅	C₅×C₅⋊C₈	C₅²⋊₃C₈	Dic₅⋊₂D₅	C₅×Dic₅	C₂×C₅⋊D₅	C₅⋊C₈	C₅²	Dic₅	C₁₀	C₅	Dic₅	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	2	2	2	4	8	1	1	2	2	2

Matrix representation of Dic₅.F₅ ►in GL₆(𝔽₄₁)

35	1	0	0	0	0
40	0	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	1

32	0	0	0	0	0
28	9	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	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	1	0	0
0	0	40	0	0	1
0	0	40	0	0	0
0	0	40	0	1	0

13	23	0	0	0	0
18	28	0	0	0	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,28,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[13,18,0,0,0,0,23,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;

Dic₅.F₅ in GAP, Magma, Sage, TeX

{\rm Dic}_5.F_5

% in TeX

G:=Group("Dic5.F5");

// GroupNames label

G:=SmallGroup(400,123);

// by ID

G=gap.SmallGroup(400,123);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,970,5765,2897]);

// Polycyclic

G:=Group<a,b,c,d|a^10=c^5=1,b^2=d^4=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;

// generators/relations

Export

Subgroup lattice of Dic₅.F₅ in TeX

G = Dic5.F5 order 400 = 24·52

1st non-split extension by Dic5 of F5 acting via F5/D5=C2

G = Dic₅.F₅ order 400 = 2⁴·5²

1^st non-split extension by Dic₅ of F₅ acting via F₅/D₅=C₂