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## G = C60⋊S3order 360 = 23·32·5

### 1st semidirect product of C60 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C60⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C2×C3⋊D15 — C60⋊S3
 Lower central C3×C15 — C3×C30 — C60⋊S3
 Upper central C1 — C2 — C4

Generators and relations for C60⋊S3
G = < a,b,c | a60=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 960 in 96 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, D4, C32, D5, C10, C12, D6, C15, C3⋊S3, C3×C6, C20, D10, D12, D15, C30, C3×C12, C2×C3⋊S3, D20, C3×C15, C60, D30, C12⋊S3, C3⋊D15, C3×C30, D60, C3×C60, C2×C3⋊D15, C60⋊S3
Quotients: C1, C2, C22, S3, D4, D5, D6, C3⋊S3, D10, D12, D15, C2×C3⋊S3, D20, D30, C12⋊S3, C3⋊D15, D60, C2×C3⋊D15, C60⋊S3

Smallest permutation representation of C60⋊S3
On 180 points
Generators in S180
```(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)
(1 151 117)(2 152 118)(3 153 119)(4 154 120)(5 155 61)(6 156 62)(7 157 63)(8 158 64)(9 159 65)(10 160 66)(11 161 67)(12 162 68)(13 163 69)(14 164 70)(15 165 71)(16 166 72)(17 167 73)(18 168 74)(19 169 75)(20 170 76)(21 171 77)(22 172 78)(23 173 79)(24 174 80)(25 175 81)(26 176 82)(27 177 83)(28 178 84)(29 179 85)(30 180 86)(31 121 87)(32 122 88)(33 123 89)(34 124 90)(35 125 91)(36 126 92)(37 127 93)(38 128 94)(39 129 95)(40 130 96)(41 131 97)(42 132 98)(43 133 99)(44 134 100)(45 135 101)(46 136 102)(47 137 103)(48 138 104)(49 139 105)(50 140 106)(51 141 107)(52 142 108)(53 143 109)(54 144 110)(55 145 111)(56 146 112)(57 147 113)(58 148 114)(59 149 115)(60 150 116)
(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(81 127)(82 126)(83 125)(84 124)(85 123)(86 122)(87 121)(88 180)(89 179)(90 178)(91 177)(92 176)(93 175)(94 174)(95 173)(96 172)(97 171)(98 170)(99 169)(100 168)(101 167)(102 166)(103 165)(104 164)(105 163)(106 162)(107 161)(108 160)(109 159)(110 158)(111 157)(112 156)(113 155)(114 154)(115 153)(116 152)(117 151)(118 150)(119 149)(120 148)```

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

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

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

93 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 5A 5B 6A 6B 6C 6D 10A 10B 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 4 5 5 6 6 6 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 90 90 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

93 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 D4 D5 D6 D10 D12 D15 D20 D30 D60 kernel C60⋊S3 C3×C60 C2×C3⋊D15 C60 C3×C15 C3×C12 C30 C3×C6 C15 C12 C32 C6 C3 # reps 1 1 2 4 1 2 4 2 8 16 4 16 32

Matrix representation of C60⋊S3 in GL4(𝔽61) generated by

 38 5 0 0 56 9 0 0 0 0 0 7 0 0 26 8
,
 1 0 0 0 0 1 0 0 0 0 59 56 0 0 25 1
,
 53 16 0 0 38 8 0 0 0 0 1 0 0 0 36 60
`G:=sub<GL(4,GF(61))| [38,56,0,0,5,9,0,0,0,0,0,26,0,0,7,8],[1,0,0,0,0,1,0,0,0,0,59,25,0,0,56,1],[53,38,0,0,16,8,0,0,0,0,1,36,0,0,0,60] >;`

C60⋊S3 in GAP, Magma, Sage, TeX

`C_{60}\rtimes S_3`
`% in TeX`

`G:=Group("C60:S3");`
`// GroupNames label`

`G:=SmallGroup(360,112);`
`// by ID`

`G=gap.SmallGroup(360,112);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,387,1444,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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