C3xDic15 - GroupNames

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

C₃×Dic₁₅ is a maximal subgroup of
C₃×D₅×Dic₃ C₃×S₃×Dic₅ C₆.D₃₀ D₆⋊₂D₁₅ C₃⋊Dic₃₀ D₃₀.S₃ Dic₁₅⋊S₃ D₃₀⋊S₃ C₃²⋊₃Dic₁₀ C₁₂×D₁₅

54 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	4A	4B	5A	5B	6A	6B	6C	6D	6E	10A	10B	12A	12B	12C	12D	15A	···	15P	30A	···	30P
order	1	2	3	3	3	3	3	4	4	5	5	6	6	6	6	6	10	10	12	12	12	12	15	···	15	30	···	30
size	1	1	1	1	2	2	2	15	15	2	2	1	1	2	2	2	2	2	15	15	15	15	2	···	2	2	···	2

54 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	C₃	C₄	C₆	C₁₂	S₃	D₅	Dic₃	C₃×S₃	Dic₅	C₃×D₅	D₁₅	C₃×Dic₃	C₃×Dic₅	Dic₁₅	C₃×D₁₅	C₃×Dic₁₅
kernel	C₃×Dic₁₅	C₃×C₃₀	Dic₁₅	C₃×C₁₅	C₃₀	C₁₅	C₃₀	C₃×C₆	C₁₅	C₁₀	C₃²	C₆	C₆	C₅	C₃	C₃	C₂	C₁
# reps	1	1	2	2	2	4	1	2	1	2	2	4	4	2	4	4	8	8

Matrix representation of C₃×Dic₁₅ ►in GL₂(𝔽₃₁) generated by

5	0
0	5

13	0
0	12

0	30
1	0

G:=sub<GL(2,GF(31))| [5,0,0,5],[13,0,0,12],[0,1,30,0] >;

C₃×Dic₁₅ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{15}

% in TeX

G:=Group("C3xDic15");

// GroupNames label

G:=SmallGroup(180,15);

// by ID

G=gap.SmallGroup(180,15);

# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,483,3604]);

// Polycyclic

G:=Group<a,b,c|a^3=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 C₃×Dic₁₅ in TeX

G = C3×Dic15 order 180 = 22·32·5

Direct product of C3 and Dic15

G = C₃×Dic₁₅ order 180 = 2²·3²·5

Direct product of C₃ and Dic₁₅