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G = S3×C65order 390 = 2·3·5·13

Direct product of C65 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: S3×C65, C3⋊C130, C1957C2, C393C10, C153C26, SmallGroup(390,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C65
C1C3C39C195 — S3×C65
C3 — S3×C65
C1C65

Generators and relations for S3×C65
 G = < a,b,c | a65=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C10
3C26
3C130

Smallest permutation representation of S3×C65
On 195 points
Generators in S195
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)
(1 85 149)(2 86 150)(3 87 151)(4 88 152)(5 89 153)(6 90 154)(7 91 155)(8 92 156)(9 93 157)(10 94 158)(11 95 159)(12 96 160)(13 97 161)(14 98 162)(15 99 163)(16 100 164)(17 101 165)(18 102 166)(19 103 167)(20 104 168)(21 105 169)(22 106 170)(23 107 171)(24 108 172)(25 109 173)(26 110 174)(27 111 175)(28 112 176)(29 113 177)(30 114 178)(31 115 179)(32 116 180)(33 117 181)(34 118 182)(35 119 183)(36 120 184)(37 121 185)(38 122 186)(39 123 187)(40 124 188)(41 125 189)(42 126 190)(43 127 191)(44 128 192)(45 129 193)(46 130 194)(47 66 195)(48 67 131)(49 68 132)(50 69 133)(51 70 134)(52 71 135)(53 72 136)(54 73 137)(55 74 138)(56 75 139)(57 76 140)(58 77 141)(59 78 142)(60 79 143)(61 80 144)(62 81 145)(63 82 146)(64 83 147)(65 84 148)
(66 195)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 161)(98 162)(99 163)(100 164)(101 165)(102 166)(103 167)(104 168)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 181)(118 182)(119 183)(120 184)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)

G:=sub<Sym(195)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,85,149)(2,86,150)(3,87,151)(4,88,152)(5,89,153)(6,90,154)(7,91,155)(8,92,156)(9,93,157)(10,94,158)(11,95,159)(12,96,160)(13,97,161)(14,98,162)(15,99,163)(16,100,164)(17,101,165)(18,102,166)(19,103,167)(20,104,168)(21,105,169)(22,106,170)(23,107,171)(24,108,172)(25,109,173)(26,110,174)(27,111,175)(28,112,176)(29,113,177)(30,114,178)(31,115,179)(32,116,180)(33,117,181)(34,118,182)(35,119,183)(36,120,184)(37,121,185)(38,122,186)(39,123,187)(40,124,188)(41,125,189)(42,126,190)(43,127,191)(44,128,192)(45,129,193)(46,130,194)(47,66,195)(48,67,131)(49,68,132)(50,69,133)(51,70,134)(52,71,135)(53,72,136)(54,73,137)(55,74,138)(56,75,139)(57,76,140)(58,77,141)(59,78,142)(60,79,143)(61,80,144)(62,81,145)(63,82,146)(64,83,147)(65,84,148), (66,195)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,163)(100,164)(101,165)(102,166)(103,167)(104,168)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194)>;

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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,85,149)(2,86,150)(3,87,151)(4,88,152)(5,89,153)(6,90,154)(7,91,155)(8,92,156)(9,93,157)(10,94,158)(11,95,159)(12,96,160)(13,97,161)(14,98,162)(15,99,163)(16,100,164)(17,101,165)(18,102,166)(19,103,167)(20,104,168)(21,105,169)(22,106,170)(23,107,171)(24,108,172)(25,109,173)(26,110,174)(27,111,175)(28,112,176)(29,113,177)(30,114,178)(31,115,179)(32,116,180)(33,117,181)(34,118,182)(35,119,183)(36,120,184)(37,121,185)(38,122,186)(39,123,187)(40,124,188)(41,125,189)(42,126,190)(43,127,191)(44,128,192)(45,129,193)(46,130,194)(47,66,195)(48,67,131)(49,68,132)(50,69,133)(51,70,134)(52,71,135)(53,72,136)(54,73,137)(55,74,138)(56,75,139)(57,76,140)(58,77,141)(59,78,142)(60,79,143)(61,80,144)(62,81,145)(63,82,146)(64,83,147)(65,84,148), (66,195)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,163)(100,164)(101,165)(102,166)(103,167)(104,168)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194) );

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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)], [(1,85,149),(2,86,150),(3,87,151),(4,88,152),(5,89,153),(6,90,154),(7,91,155),(8,92,156),(9,93,157),(10,94,158),(11,95,159),(12,96,160),(13,97,161),(14,98,162),(15,99,163),(16,100,164),(17,101,165),(18,102,166),(19,103,167),(20,104,168),(21,105,169),(22,106,170),(23,107,171),(24,108,172),(25,109,173),(26,110,174),(27,111,175),(28,112,176),(29,113,177),(30,114,178),(31,115,179),(32,116,180),(33,117,181),(34,118,182),(35,119,183),(36,120,184),(37,121,185),(38,122,186),(39,123,187),(40,124,188),(41,125,189),(42,126,190),(43,127,191),(44,128,192),(45,129,193),(46,130,194),(47,66,195),(48,67,131),(49,68,132),(50,69,133),(51,70,134),(52,71,135),(53,72,136),(54,73,137),(55,74,138),(56,75,139),(57,76,140),(58,77,141),(59,78,142),(60,79,143),(61,80,144),(62,81,145),(63,82,146),(64,83,147),(65,84,148)], [(66,195),(67,131),(68,132),(69,133),(70,134),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(81,145),(82,146),(83,147),(84,148),(85,149),(86,150),(87,151),(88,152),(89,153),(90,154),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,161),(98,162),(99,163),(100,164),(101,165),(102,166),(103,167),(104,168),(105,169),(106,170),(107,171),(108,172),(109,173),(110,174),(111,175),(112,176),(113,177),(114,178),(115,179),(116,180),(117,181),(118,182),(119,183),(120,184),(121,185),(122,186),(123,187),(124,188),(125,189),(126,190),(127,191),(128,192),(129,193),(130,194)])

195 conjugacy classes

class 1  2  3 5A5B5C5D10A10B10C10D13A···13L15A15B15C15D26A···26L39A···39L65A···65AV130A···130AV195A···195AV
order12355551010101013···131515151526···2639···3965···65130···130195···195
size132111133331···122223···32···21···13···32···2

195 irreducible representations

dim111111112222
type+++
imageC1C2C5C10C13C26C65C130S3C5×S3S3×C13S3×C65
kernelS3×C65C195S3×C13C39C5×S3C15S3C3C65C13C5C1
# reps114412124848141248

Matrix representation of S3×C65 in GL2(𝔽1171) generated by

10090
01009
,
01170
11170
,
11170
01170
G:=sub<GL(2,GF(1171))| [1009,0,0,1009],[0,1,1170,1170],[1,0,1170,1170] >;

S3×C65 in GAP, Magma, Sage, TeX

S_3\times C_{65}
% in TeX

G:=Group("S3xC65");
// GroupNames label

G:=SmallGroup(390,8);
// by ID

G=gap.SmallGroup(390,8);
# by ID

G:=PCGroup([4,-2,-5,-13,-3,4163]);
// Polycyclic

G:=Group<a,b,c|a^65=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C65 in TeX

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