C5xDic15 - GroupNames

(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 37 43 49 55)(32 38 44 50 56)(33 39 45 51 57)(34 40 46 52 58)(35 41 47 53 59)(36 42 48 54 60)

(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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)

(1 36 16 51)(2 35 17 50)(3 34 18 49)(4 33 19 48)(5 32 20 47)(6 31 21 46)(7 60 22 45)(8 59 23 44)(9 58 24 43)(10 57 25 42)(11 56 26 41)(12 55 27 40)(13 54 28 39)(14 53 29 38)(15 52 30 37)

G:=sub<Sym(60)| (1,25,19,13,7)(2,26,20,14,8)(3,27,21,15,9)(4,28,22,16,10)(5,29,23,17,11)(6,30,24,18,12)(31,37,43,49,55)(32,38,44,50,56)(33,39,45,51,57)(34,40,46,52,58)(35,41,47,53,59)(36,42,48,54,60), (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,36,16,51)(2,35,17,50)(3,34,18,49)(4,33,19,48)(5,32,20,47)(6,31,21,46)(7,60,22,45)(8,59,23,44)(9,58,24,43)(10,57,25,42)(11,56,26,41)(12,55,27,40)(13,54,28,39)(14,53,29,38)(15,52,30,37)>;

G:=Group( (1,25,19,13,7)(2,26,20,14,8)(3,27,21,15,9)(4,28,22,16,10)(5,29,23,17,11)(6,30,24,18,12)(31,37,43,49,55)(32,38,44,50,56)(33,39,45,51,57)(34,40,46,52,58)(35,41,47,53,59)(36,42,48,54,60), (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,36,16,51)(2,35,17,50)(3,34,18,49)(4,33,19,48)(5,32,20,47)(6,31,21,46)(7,60,22,45)(8,59,23,44)(9,58,24,43)(10,57,25,42)(11,56,26,41)(12,55,27,40)(13,54,28,39)(14,53,29,38)(15,52,30,37) );

G=PermutationGroup([[(1,25,19,13,7),(2,26,20,14,8),(3,27,21,15,9),(4,28,22,16,10),(5,29,23,17,11),(6,30,24,18,12),(31,37,43,49,55),(32,38,44,50,56),(33,39,45,51,57),(34,40,46,52,58),(35,41,47,53,59),(36,42,48,54,60)], [(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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,36,16,51),(2,35,17,50),(3,34,18,49),(4,33,19,48),(5,32,20,47),(6,31,21,46),(7,60,22,45),(8,59,23,44),(9,58,24,43),(10,57,25,42),(11,56,26,41),(12,55,27,40),(13,54,28,39),(14,53,29,38),(15,52,30,37)]])

90 conjugacy classes

class	1	2	3	4A	4B	5A	5B	5C	5D	5E	···	5N	6	10A	10B	10C	10D	10E	···	10N	15A	···	15X	20A	···	20H	30A	···	30X
order	1	2	3	4	4	5	5	5	5	5	···	5	6	10	10	10	10	10	···	10	15	···	15	20	···	20	30	···	30
size	1	1	2	15	15	1	1	1	1	2	···	2	2	1	1	1	1	2	···	2	2	···	2	15	···	15	2	···	2

90 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+					+	+	-	-		+			-
image	C₁	C₂	C₄	C₅	C₁₀	C₂₀	S₃	D₅	Dic₃	Dic₅	C₅×S₃	D₁₅	C₅×D₅	C₅×Dic₃	Dic₁₅	C₅×Dic₅	C₅×D₁₅	C₅×Dic₁₅
kernel	C₅×Dic₁₅	C₅×C₃₀	C₅×C₁₅	Dic₁₅	C₃₀	C₁₅	C₅×C₁₀	C₃₀	C₅²	C₁₅	C₁₀	C₁₀	C₆	C₅	C₅	C₃	C₂	C₁
# reps	1	1	2	4	4	8	1	2	1	2	4	4	8	4	4	8	16	16

Matrix representation of C₅×Dic₁₅ ►in GL₂(𝔽₃₁) generated by

2	0
0	2

17	0
0	11

0	30
1	0

G:=sub<GL(2,GF(31))| [2,0,0,2],[17,0,0,11],[0,1,30,0] >;

C₅×Dic₁₅ in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{15}

% in TeX

G:=Group("C5xDic15");

// GroupNames label

G:=SmallGroup(300,19);

// by ID

G=gap.SmallGroup(300,19);

# by ID

G:=PCGroup([5,-2,-5,-2,-3,-5,50,803,6004]);

// Polycyclic

G:=Group<a,b,c|a^5=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of C₅×Dic₁₅ in TeX

G = C5×Dic15 order 300 = 22·3·52

Direct product of C5 and Dic15

G = C₅×Dic₁₅ order 300 = 2²·3·5²

Direct product of C₅ and Dic₁₅