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## G = C15×Dic6order 360 = 23·32·5

### Direct product of C15 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C15×Dic6
 Chief series C1 — C3 — C6 — C30 — C3×C30 — Dic3×C15 — C15×Dic6
 Lower central C3 — C6 — C15×Dic6
 Upper central C1 — C30 — C60

Generators and relations for C15×Dic6
G = < a,b,c | a15=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 84 in 54 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C5, C6, C6, Q8, C32, C10, Dic3, C12, C12, C15, C15, C3×C6, C20, C20, Dic6, C3×Q8, C30, C30, C3×Dic3, C3×C12, C5×Q8, C3×C15, C5×Dic3, C60, C60, C3×Dic6, C3×C30, C5×Dic6, Q8×C15, Dic3×C15, C3×C60, C15×Dic6
Quotients: C1, C2, C3, C22, C5, S3, C6, Q8, C10, D6, C2×C6, C15, C3×S3, C2×C10, Dic6, C3×Q8, C5×S3, C30, S3×C6, C5×Q8, S3×C10, C2×C30, C3×Dic6, S3×C15, C5×Dic6, Q8×C15, S3×C30, C15×Dic6

Smallest permutation representation of C15×Dic6
On 120 points
Generators in S120
(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)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 18 75 99 6 23 65 104 11 28 70 94)(2 19 61 100 7 24 66 105 12 29 71 95)(3 20 62 101 8 25 67 91 13 30 72 96)(4 21 63 102 9 26 68 92 14 16 73 97)(5 22 64 103 10 27 69 93 15 17 74 98)(31 56 77 110 41 51 87 120 36 46 82 115)(32 57 78 111 42 52 88 106 37 47 83 116)(33 58 79 112 43 53 89 107 38 48 84 117)(34 59 80 113 44 54 90 108 39 49 85 118)(35 60 81 114 45 55 76 109 40 50 86 119)
(1 40 65 81)(2 41 66 82)(3 42 67 83)(4 43 68 84)(5 44 69 85)(6 45 70 86)(7 31 71 87)(8 32 72 88)(9 33 73 89)(10 34 74 90)(11 35 75 76)(12 36 61 77)(13 37 62 78)(14 38 63 79)(15 39 64 80)(16 107 102 58)(17 108 103 59)(18 109 104 60)(19 110 105 46)(20 111 91 47)(21 112 92 48)(22 113 93 49)(23 114 94 50)(24 115 95 51)(25 116 96 52)(26 117 97 53)(27 118 98 54)(28 119 99 55)(29 120 100 56)(30 106 101 57)

G:=sub<Sym(120)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,18,75,99,6,23,65,104,11,28,70,94)(2,19,61,100,7,24,66,105,12,29,71,95)(3,20,62,101,8,25,67,91,13,30,72,96)(4,21,63,102,9,26,68,92,14,16,73,97)(5,22,64,103,10,27,69,93,15,17,74,98)(31,56,77,110,41,51,87,120,36,46,82,115)(32,57,78,111,42,52,88,106,37,47,83,116)(33,58,79,112,43,53,89,107,38,48,84,117)(34,59,80,113,44,54,90,108,39,49,85,118)(35,60,81,114,45,55,76,109,40,50,86,119), (1,40,65,81)(2,41,66,82)(3,42,67,83)(4,43,68,84)(5,44,69,85)(6,45,70,86)(7,31,71,87)(8,32,72,88)(9,33,73,89)(10,34,74,90)(11,35,75,76)(12,36,61,77)(13,37,62,78)(14,38,63,79)(15,39,64,80)(16,107,102,58)(17,108,103,59)(18,109,104,60)(19,110,105,46)(20,111,91,47)(21,112,92,48)(22,113,93,49)(23,114,94,50)(24,115,95,51)(25,116,96,52)(26,117,97,53)(27,118,98,54)(28,119,99,55)(29,120,100,56)(30,106,101,57)>;

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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,18,75,99,6,23,65,104,11,28,70,94)(2,19,61,100,7,24,66,105,12,29,71,95)(3,20,62,101,8,25,67,91,13,30,72,96)(4,21,63,102,9,26,68,92,14,16,73,97)(5,22,64,103,10,27,69,93,15,17,74,98)(31,56,77,110,41,51,87,120,36,46,82,115)(32,57,78,111,42,52,88,106,37,47,83,116)(33,58,79,112,43,53,89,107,38,48,84,117)(34,59,80,113,44,54,90,108,39,49,85,118)(35,60,81,114,45,55,76,109,40,50,86,119), (1,40,65,81)(2,41,66,82)(3,42,67,83)(4,43,68,84)(5,44,69,85)(6,45,70,86)(7,31,71,87)(8,32,72,88)(9,33,73,89)(10,34,74,90)(11,35,75,76)(12,36,61,77)(13,37,62,78)(14,38,63,79)(15,39,64,80)(16,107,102,58)(17,108,103,59)(18,109,104,60)(19,110,105,46)(20,111,91,47)(21,112,92,48)(22,113,93,49)(23,114,94,50)(24,115,95,51)(25,116,96,52)(26,117,97,53)(27,118,98,54)(28,119,99,55)(29,120,100,56)(30,106,101,57) );

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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,18,75,99,6,23,65,104,11,28,70,94),(2,19,61,100,7,24,66,105,12,29,71,95),(3,20,62,101,8,25,67,91,13,30,72,96),(4,21,63,102,9,26,68,92,14,16,73,97),(5,22,64,103,10,27,69,93,15,17,74,98),(31,56,77,110,41,51,87,120,36,46,82,115),(32,57,78,111,42,52,88,106,37,47,83,116),(33,58,79,112,43,53,89,107,38,48,84,117),(34,59,80,113,44,54,90,108,39,49,85,118),(35,60,81,114,45,55,76,109,40,50,86,119)], [(1,40,65,81),(2,41,66,82),(3,42,67,83),(4,43,68,84),(5,44,69,85),(6,45,70,86),(7,31,71,87),(8,32,72,88),(9,33,73,89),(10,34,74,90),(11,35,75,76),(12,36,61,77),(13,37,62,78),(14,38,63,79),(15,39,64,80),(16,107,102,58),(17,108,103,59),(18,109,104,60),(19,110,105,46),(20,111,91,47),(21,112,92,48),(22,113,93,49),(23,114,94,50),(24,115,95,51),(25,116,96,52),(26,117,97,53),(27,118,98,54),(28,119,99,55),(29,120,100,56),(30,106,101,57)]])

135 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A 10B 10C 10D 12A ··· 12H 12I 12J 12K 12L 15A ··· 15H 15I ··· 15T 20A 20B 20C 20D 20E ··· 20L 30A ··· 30H 30I ··· 30T 60A ··· 60AF 60AG ··· 60AV order 1 2 3 3 3 3 3 4 4 4 5 5 5 5 6 6 6 6 6 10 10 10 10 12 ··· 12 12 12 12 12 15 ··· 15 15 ··· 15 20 20 20 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 1 1 2 2 2 2 6 6 1 1 1 1 1 1 2 2 2 1 1 1 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 S3 Q8 D6 C3×S3 Dic6 C3×Q8 C5×S3 S3×C6 C5×Q8 S3×C10 C3×Dic6 S3×C15 C5×Dic6 Q8×C15 S3×C30 C15×Dic6 kernel C15×Dic6 Dic3×C15 C3×C60 C5×Dic6 C3×Dic6 C5×Dic3 C60 C3×Dic3 C3×C12 Dic6 Dic3 C12 C60 C3×C15 C30 C20 C15 C15 C12 C10 C32 C6 C5 C4 C3 C3 C2 C1 # reps 1 2 1 2 4 4 2 8 4 8 16 8 1 1 1 2 2 2 4 2 4 4 4 8 8 8 8 16

Matrix representation of C15×Dic6 in GL2(𝔽61) generated by

 57 0 0 57
,
 40 0 0 29
,
 0 1 60 0
G:=sub<GL(2,GF(61))| [57,0,0,57],[40,0,0,29],[0,60,1,0] >;

C15×Dic6 in GAP, Magma, Sage, TeX

C_{15}\times {\rm Dic}_6
% in TeX

G:=Group("C15xDic6");
// GroupNames label

G:=SmallGroup(360,95);
// by ID

G=gap.SmallGroup(360,95);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,360,745,367,8645]);
// Polycyclic

G:=Group<a,b,c|a^15=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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