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## G = C3⋊Dic15order 180 = 22·32·5

### The semidirect product of C3 and Dic15 acting via Dic15/C30=C2

Aliases: C3⋊Dic15, C30.3S3, C6.3D15, C153Dic3, C323Dic5, (C3×C15)⋊7C4, C10.(C3⋊S3), C2.(C3⋊D15), (C3×C6).2D5, (C3×C30).1C2, C52(C3⋊Dic3), SmallGroup(180,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C3⋊Dic15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3⋊Dic15
 Lower central C3×C15 — C3⋊Dic15
 Upper central C1 — C2

Generators and relations for C3⋊Dic15
G = < a,b,c | a3=b30=1, c2=b15, ab=ba, cac-1=a-1, cbc-1=b-1 >

Smallest permutation representation of C3⋊Dic15
Regular action on 180 points
Generators in S180
(1 67 134)(2 68 135)(3 69 136)(4 70 137)(5 71 138)(6 72 139)(7 73 140)(8 74 141)(9 75 142)(10 76 143)(11 77 144)(12 78 145)(13 79 146)(14 80 147)(15 81 148)(16 82 149)(17 83 150)(18 84 121)(19 85 122)(20 86 123)(21 87 124)(22 88 125)(23 89 126)(24 90 127)(25 61 128)(26 62 129)(27 63 130)(28 64 131)(29 65 132)(30 66 133)(31 180 100)(32 151 101)(33 152 102)(34 153 103)(35 154 104)(36 155 105)(37 156 106)(38 157 107)(39 158 108)(40 159 109)(41 160 110)(42 161 111)(43 162 112)(44 163 113)(45 164 114)(46 165 115)(47 166 116)(48 167 117)(49 168 118)(50 169 119)(51 170 120)(52 171 91)(53 172 92)(54 173 93)(55 174 94)(56 175 95)(57 176 96)(58 177 97)(59 178 98)(60 179 99)
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
(1 37 16 52)(2 36 17 51)(3 35 18 50)(4 34 19 49)(5 33 20 48)(6 32 21 47)(7 31 22 46)(8 60 23 45)(9 59 24 44)(10 58 25 43)(11 57 26 42)(12 56 27 41)(13 55 28 40)(14 54 29 39)(15 53 30 38)(61 112 76 97)(62 111 77 96)(63 110 78 95)(64 109 79 94)(65 108 80 93)(66 107 81 92)(67 106 82 91)(68 105 83 120)(69 104 84 119)(70 103 85 118)(71 102 86 117)(72 101 87 116)(73 100 88 115)(74 99 89 114)(75 98 90 113)(121 169 136 154)(122 168 137 153)(123 167 138 152)(124 166 139 151)(125 165 140 180)(126 164 141 179)(127 163 142 178)(128 162 143 177)(129 161 144 176)(130 160 145 175)(131 159 146 174)(132 158 147 173)(133 157 148 172)(134 156 149 171)(135 155 150 170)

G:=sub<Sym(180)| (1,67,134)(2,68,135)(3,69,136)(4,70,137)(5,71,138)(6,72,139)(7,73,140)(8,74,141)(9,75,142)(10,76,143)(11,77,144)(12,78,145)(13,79,146)(14,80,147)(15,81,148)(16,82,149)(17,83,150)(18,84,121)(19,85,122)(20,86,123)(21,87,124)(22,88,125)(23,89,126)(24,90,127)(25,61,128)(26,62,129)(27,63,130)(28,64,131)(29,65,132)(30,66,133)(31,180,100)(32,151,101)(33,152,102)(34,153,103)(35,154,104)(36,155,105)(37,156,106)(38,157,107)(39,158,108)(40,159,109)(41,160,110)(42,161,111)(43,162,112)(44,163,113)(45,164,114)(46,165,115)(47,166,116)(48,167,117)(49,168,118)(50,169,119)(51,170,120)(52,171,91)(53,172,92)(54,173,93)(55,174,94)(56,175,95)(57,176,96)(58,177,97)(59,178,98)(60,179,99), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,37,16,52)(2,36,17,51)(3,35,18,50)(4,34,19,49)(5,33,20,48)(6,32,21,47)(7,31,22,46)(8,60,23,45)(9,59,24,44)(10,58,25,43)(11,57,26,42)(12,56,27,41)(13,55,28,40)(14,54,29,39)(15,53,30,38)(61,112,76,97)(62,111,77,96)(63,110,78,95)(64,109,79,94)(65,108,80,93)(66,107,81,92)(67,106,82,91)(68,105,83,120)(69,104,84,119)(70,103,85,118)(71,102,86,117)(72,101,87,116)(73,100,88,115)(74,99,89,114)(75,98,90,113)(121,169,136,154)(122,168,137,153)(123,167,138,152)(124,166,139,151)(125,165,140,180)(126,164,141,179)(127,163,142,178)(128,162,143,177)(129,161,144,176)(130,160,145,175)(131,159,146,174)(132,158,147,173)(133,157,148,172)(134,156,149,171)(135,155,150,170)>;

G:=Group( (1,67,134)(2,68,135)(3,69,136)(4,70,137)(5,71,138)(6,72,139)(7,73,140)(8,74,141)(9,75,142)(10,76,143)(11,77,144)(12,78,145)(13,79,146)(14,80,147)(15,81,148)(16,82,149)(17,83,150)(18,84,121)(19,85,122)(20,86,123)(21,87,124)(22,88,125)(23,89,126)(24,90,127)(25,61,128)(26,62,129)(27,63,130)(28,64,131)(29,65,132)(30,66,133)(31,180,100)(32,151,101)(33,152,102)(34,153,103)(35,154,104)(36,155,105)(37,156,106)(38,157,107)(39,158,108)(40,159,109)(41,160,110)(42,161,111)(43,162,112)(44,163,113)(45,164,114)(46,165,115)(47,166,116)(48,167,117)(49,168,118)(50,169,119)(51,170,120)(52,171,91)(53,172,92)(54,173,93)(55,174,94)(56,175,95)(57,176,96)(58,177,97)(59,178,98)(60,179,99), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,37,16,52)(2,36,17,51)(3,35,18,50)(4,34,19,49)(5,33,20,48)(6,32,21,47)(7,31,22,46)(8,60,23,45)(9,59,24,44)(10,58,25,43)(11,57,26,42)(12,56,27,41)(13,55,28,40)(14,54,29,39)(15,53,30,38)(61,112,76,97)(62,111,77,96)(63,110,78,95)(64,109,79,94)(65,108,80,93)(66,107,81,92)(67,106,82,91)(68,105,83,120)(69,104,84,119)(70,103,85,118)(71,102,86,117)(72,101,87,116)(73,100,88,115)(74,99,89,114)(75,98,90,113)(121,169,136,154)(122,168,137,153)(123,167,138,152)(124,166,139,151)(125,165,140,180)(126,164,141,179)(127,163,142,178)(128,162,143,177)(129,161,144,176)(130,160,145,175)(131,159,146,174)(132,158,147,173)(133,157,148,172)(134,156,149,171)(135,155,150,170) );

G=PermutationGroup([[(1,67,134),(2,68,135),(3,69,136),(4,70,137),(5,71,138),(6,72,139),(7,73,140),(8,74,141),(9,75,142),(10,76,143),(11,77,144),(12,78,145),(13,79,146),(14,80,147),(15,81,148),(16,82,149),(17,83,150),(18,84,121),(19,85,122),(20,86,123),(21,87,124),(22,88,125),(23,89,126),(24,90,127),(25,61,128),(26,62,129),(27,63,130),(28,64,131),(29,65,132),(30,66,133),(31,180,100),(32,151,101),(33,152,102),(34,153,103),(35,154,104),(36,155,105),(37,156,106),(38,157,107),(39,158,108),(40,159,109),(41,160,110),(42,161,111),(43,162,112),(44,163,113),(45,164,114),(46,165,115),(47,166,116),(48,167,117),(49,168,118),(50,169,119),(51,170,120),(52,171,91),(53,172,92),(54,173,93),(55,174,94),(56,175,95),(57,176,96),(58,177,97),(59,178,98),(60,179,99)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)], [(1,37,16,52),(2,36,17,51),(3,35,18,50),(4,34,19,49),(5,33,20,48),(6,32,21,47),(7,31,22,46),(8,60,23,45),(9,59,24,44),(10,58,25,43),(11,57,26,42),(12,56,27,41),(13,55,28,40),(14,54,29,39),(15,53,30,38),(61,112,76,97),(62,111,77,96),(63,110,78,95),(64,109,79,94),(65,108,80,93),(66,107,81,92),(67,106,82,91),(68,105,83,120),(69,104,84,119),(70,103,85,118),(71,102,86,117),(72,101,87,116),(73,100,88,115),(74,99,89,114),(75,98,90,113),(121,169,136,154),(122,168,137,153),(123,167,138,152),(124,166,139,151),(125,165,140,180),(126,164,141,179),(127,163,142,178),(128,162,143,177),(129,161,144,176),(130,160,145,175),(131,159,146,174),(132,158,147,173),(133,157,148,172),(134,156,149,171),(135,155,150,170)]])

C3⋊Dic15 is a maximal subgroup of
(C3×C6).F5  D5×C3⋊Dic3  C3⋊S3×Dic5  C30.12D6  C15⋊Dic6  Dic3×D15  S3×Dic15  D6⋊D15  C3⋊Dic30  C12.D15  C4×C3⋊D15  C62⋊D5
C3⋊Dic15 is a maximal quotient of
C60.S3

48 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 5A 5B 6A 6B 6C 6D 10A 10B 15A ··· 15P 30A ··· 30P order 1 2 3 3 3 3 4 4 5 5 6 6 6 6 10 10 15 ··· 15 30 ··· 30 size 1 1 2 2 2 2 45 45 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C4 S3 D5 Dic3 Dic5 D15 Dic15 kernel C3⋊Dic15 C3×C30 C3×C15 C30 C3×C6 C15 C32 C6 C3 # reps 1 1 2 4 2 4 2 16 16

Matrix representation of C3⋊Dic15 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 52 28 0 0 17 8
,
 45 9 0 0 52 5 0 0 0 0 56 25 0 0 13 8
,
 58 5 0 0 59 3 0 0 0 0 3 27 0 0 29 58
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,52,17,0,0,28,8],[45,52,0,0,9,5,0,0,0,0,56,13,0,0,25,8],[58,59,0,0,5,3,0,0,0,0,3,29,0,0,27,58] >;

C3⋊Dic15 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_{15}
% in TeX

G:=Group("C3:Dic15");
// GroupNames label

G:=SmallGroup(180,17);
// by ID

G=gap.SmallGroup(180,17);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,10,122,483,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^30=1,c^2=b^15,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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