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## G = C5×C12⋊S3order 360 = 23·32·5

### Direct product of C5 and C12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C5×C12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — C10×C3⋊S3 — C5×C12⋊S3
 Lower central C32 — C3×C6 — C5×C12⋊S3
 Upper central C1 — C10 — C20

Generators and relations for C5×C12⋊S3
G = < a,b,c,d | a5=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 344 in 96 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, D4, C32, C10, C10, C12, D6, C15, C3⋊S3, C3×C6, C20, C2×C10, D12, C5×S3, C30, C3×C12, C2×C3⋊S3, C5×D4, C3×C15, C60, S3×C10, C12⋊S3, C5×C3⋊S3, C3×C30, C5×D12, C3×C60, C10×C3⋊S3, C5×C12⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C3⋊S3, C2×C10, D12, C5×S3, C2×C3⋊S3, C5×D4, S3×C10, C12⋊S3, C5×C3⋊S3, C5×D12, C10×C3⋊S3, C5×C12⋊S3

Smallest permutation representation of C5×C12⋊S3
On 180 points
Generators in S180
(1 121 52 22 90)(2 122 53 23 91)(3 123 54 24 92)(4 124 55 13 93)(5 125 56 14 94)(6 126 57 15 95)(7 127 58 16 96)(8 128 59 17 85)(9 129 60 18 86)(10 130 49 19 87)(11 131 50 20 88)(12 132 51 21 89)(25 172 112 75 158)(26 173 113 76 159)(27 174 114 77 160)(28 175 115 78 161)(29 176 116 79 162)(30 177 117 80 163)(31 178 118 81 164)(32 179 119 82 165)(33 180 120 83 166)(34 169 109 84 167)(35 170 110 73 168)(36 171 111 74 157)(37 66 139 97 149)(38 67 140 98 150)(39 68 141 99 151)(40 69 142 100 152)(41 70 143 101 153)(42 71 144 102 154)(43 72 133 103 155)(44 61 134 104 156)(45 62 135 105 145)(46 63 136 106 146)(47 64 137 107 147)(48 65 138 108 148)
(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 42 75)(2 43 76)(3 44 77)(4 45 78)(5 46 79)(6 47 80)(7 48 81)(8 37 82)(9 38 83)(10 39 84)(11 40 73)(12 41 74)(13 105 175)(14 106 176)(15 107 177)(16 108 178)(17 97 179)(18 98 180)(19 99 169)(20 100 170)(21 101 171)(22 102 172)(23 103 173)(24 104 174)(25 52 144)(26 53 133)(27 54 134)(28 55 135)(29 56 136)(30 57 137)(31 58 138)(32 59 139)(33 60 140)(34 49 141)(35 50 142)(36 51 143)(61 160 123)(62 161 124)(63 162 125)(64 163 126)(65 164 127)(66 165 128)(67 166 129)(68 167 130)(69 168 131)(70 157 132)(71 158 121)(72 159 122)(85 149 119)(86 150 120)(87 151 109)(88 152 110)(89 153 111)(90 154 112)(91 155 113)(92 156 114)(93 145 115)(94 146 116)(95 147 117)(96 148 118)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 19)(14 18)(15 17)(20 24)(21 23)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 84)(46 83)(47 82)(48 81)(49 55)(50 54)(51 53)(56 60)(57 59)(61 168)(62 167)(63 166)(64 165)(65 164)(66 163)(67 162)(68 161)(69 160)(70 159)(71 158)(72 157)(85 95)(86 94)(87 93)(88 92)(89 91)(97 177)(98 176)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 180)(107 179)(108 178)(109 145)(110 156)(111 155)(112 154)(113 153)(114 152)(115 151)(116 150)(117 149)(118 148)(119 147)(120 146)(122 132)(123 131)(124 130)(125 129)(126 128)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10L 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 4 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 18 18 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 18 ··· 18 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D4 D6 D12 C5×S3 C5×D4 S3×C10 C5×D12 kernel C5×C12⋊S3 C3×C60 C10×C3⋊S3 C12⋊S3 C3×C12 C2×C3⋊S3 C60 C3×C15 C30 C15 C12 C32 C6 C3 # reps 1 1 2 4 4 8 4 1 4 8 16 4 16 32

Matrix representation of C5×C12⋊S3 in GL4(𝔽61) generated by

 20 0 0 0 0 20 0 0 0 0 9 0 0 0 0 9
,
 0 60 0 0 1 1 0 0 0 0 53 46 0 0 45 8
,
 60 60 0 0 1 0 0 0 0 0 1 60 0 0 3 59
,
 1 0 0 0 60 60 0 0 0 0 2 60 0 0 3 59
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,60,1,0,0,0,0,53,45,0,0,46,8],[60,1,0,0,60,0,0,0,0,0,1,3,0,0,60,59],[1,60,0,0,0,60,0,0,0,0,2,3,0,0,60,59] >;

C5×C12⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_{12}\rtimes S_3
% in TeX

G:=Group("C5xC12:S3");
// GroupNames label

G:=SmallGroup(360,107);
// by ID

G=gap.SmallGroup(360,107);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,2404,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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