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G = C60⋊S3order 360 = 23·32·5

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

metabelian, supersoluble, monomial

Aliases: C601S3, C31D60, C121D15, C154D12, C325D20, C30.48D6, C6.16D30, C4⋊(C3⋊D15), (C3×C60)⋊1C2, (C3×C12)⋊1D5, (C3×C15)⋊17D4, C201(C3⋊S3), C51(C12⋊S3), (C3×C6).34D10, (C3×C30).34C22, (C2×C3⋊D15)⋊1C2, C2.4(C2×C3⋊D15), C10.10(C2×C3⋊S3), SmallGroup(360,112)

Series: Derived Chief Lower central Upper central

C1C3×C30 — C60⋊S3
C1C5C15C3×C15C3×C30C2×C3⋊D15 — C60⋊S3
C3×C15C3×C30 — C60⋊S3
C1C2C4

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 [×2], C3 [×4], C4, C22 [×2], C5, S3 [×8], C6 [×4], D4, C32, D5 [×2], C10, C12 [×4], D6 [×8], C15 [×4], C3⋊S3 [×2], C3×C6, C20, D10 [×2], D12 [×4], D15 [×8], C30 [×4], C3×C12, C2×C3⋊S3 [×2], D20, C3×C15, C60 [×4], D30 [×8], C12⋊S3, C3⋊D15 [×2], C3×C30, D60 [×4], C3×C60, C2×C3⋊D15 [×2], C60⋊S3
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D5, D6 [×4], C3⋊S3, D10, D12 [×4], D15 [×4], C2×C3⋊S3, D20, D30 [×4], C12⋊S3, C3⋊D15, D60 [×4], 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)
(1 157 76)(2 158 77)(3 159 78)(4 160 79)(5 161 80)(6 162 81)(7 163 82)(8 164 83)(9 165 84)(10 166 85)(11 167 86)(12 168 87)(13 169 88)(14 170 89)(15 171 90)(16 172 91)(17 173 92)(18 174 93)(19 175 94)(20 176 95)(21 177 96)(22 178 97)(23 179 98)(24 180 99)(25 121 100)(26 122 101)(27 123 102)(28 124 103)(29 125 104)(30 126 105)(31 127 106)(32 128 107)(33 129 108)(34 130 109)(35 131 110)(36 132 111)(37 133 112)(38 134 113)(39 135 114)(40 136 115)(41 137 116)(42 138 117)(43 139 118)(44 140 119)(45 141 120)(46 142 61)(47 143 62)(48 144 63)(49 145 64)(50 146 65)(51 147 66)(52 148 67)(53 149 68)(54 150 69)(55 151 70)(56 152 71)(57 153 72)(58 154 73)(59 155 74)(60 156 75)
(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)(61 172)(62 171)(63 170)(64 169)(65 168)(66 167)(67 166)(68 165)(69 164)(70 163)(71 162)(72 161)(73 160)(74 159)(75 158)(76 157)(77 156)(78 155)(79 154)(80 153)(81 152)(82 151)(83 150)(84 149)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 180)(114 179)(115 178)(116 177)(117 176)(118 175)(119 174)(120 173)

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

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

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

93 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 5A5B6A6B6C6D10A10B12A···12H15A···15P20A20B20C20D30A···30P60A···60AF
order122233334556666101012···1215···152020202030···3060···60
size11909022222222222222···22···222222···22···2

93 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D5D6D10D12D15D20D30D60
kernelC60⋊S3C3×C60C2×C3⋊D15C60C3×C15C3×C12C30C3×C6C15C12C32C6C3
# reps1124124281641632

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

38500
56900
0007
00268
,
1000
0100
005956
00251
,
531600
38800
0010
003660
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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