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G = S3×C73order 438 = 2·3·73

Direct product of C73 and S3

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

Aliases: S3×C73, C3⋊C146, C2193C2, SmallGroup(438,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C73
C1C3C219 — S3×C73
C3 — S3×C73
C1C73

Generators and relations for S3×C73
 G = < a,b,c | a73=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C146

Smallest permutation representation of S3×C73
On 219 points
Generators in S219
(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)(147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219)
(1 175 135)(2 176 136)(3 177 137)(4 178 138)(5 179 139)(6 180 140)(7 181 141)(8 182 142)(9 183 143)(10 184 144)(11 185 145)(12 186 146)(13 187 74)(14 188 75)(15 189 76)(16 190 77)(17 191 78)(18 192 79)(19 193 80)(20 194 81)(21 195 82)(22 196 83)(23 197 84)(24 198 85)(25 199 86)(26 200 87)(27 201 88)(28 202 89)(29 203 90)(30 204 91)(31 205 92)(32 206 93)(33 207 94)(34 208 95)(35 209 96)(36 210 97)(37 211 98)(38 212 99)(39 213 100)(40 214 101)(41 215 102)(42 216 103)(43 217 104)(44 218 105)(45 219 106)(46 147 107)(47 148 108)(48 149 109)(49 150 110)(50 151 111)(51 152 112)(52 153 113)(53 154 114)(54 155 115)(55 156 116)(56 157 117)(57 158 118)(58 159 119)(59 160 120)(60 161 121)(61 162 122)(62 163 123)(63 164 124)(64 165 125)(65 166 126)(66 167 127)(67 168 128)(68 169 129)(69 170 130)(70 171 131)(71 172 132)(72 173 133)(73 174 134)
(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(91 204)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 219)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 161)(122 162)(123 163)(124 164)(125 165)(126 166)(127 167)(128 168)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 177)(138 178)(139 179)(140 180)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)

G:=sub<Sym(219)| (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)(147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219), (1,175,135)(2,176,136)(3,177,137)(4,178,138)(5,179,139)(6,180,140)(7,181,141)(8,182,142)(9,183,143)(10,184,144)(11,185,145)(12,186,146)(13,187,74)(14,188,75)(15,189,76)(16,190,77)(17,191,78)(18,192,79)(19,193,80)(20,194,81)(21,195,82)(22,196,83)(23,197,84)(24,198,85)(25,199,86)(26,200,87)(27,201,88)(28,202,89)(29,203,90)(30,204,91)(31,205,92)(32,206,93)(33,207,94)(34,208,95)(35,209,96)(36,210,97)(37,211,98)(38,212,99)(39,213,100)(40,214,101)(41,215,102)(42,216,103)(43,217,104)(44,218,105)(45,219,106)(46,147,107)(47,148,108)(48,149,109)(49,150,110)(50,151,111)(51,152,112)(52,153,113)(53,154,114)(54,155,115)(55,156,116)(56,157,117)(57,158,118)(58,159,119)(59,160,120)(60,161,121)(61,162,122)(62,163,123)(63,164,124)(64,165,125)(65,166,126)(66,167,127)(67,168,128)(68,169,129)(69,170,130)(70,171,131)(71,172,132)(72,173,133)(73,174,134), (74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186)>;

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)(147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219), (1,175,135)(2,176,136)(3,177,137)(4,178,138)(5,179,139)(6,180,140)(7,181,141)(8,182,142)(9,183,143)(10,184,144)(11,185,145)(12,186,146)(13,187,74)(14,188,75)(15,189,76)(16,190,77)(17,191,78)(18,192,79)(19,193,80)(20,194,81)(21,195,82)(22,196,83)(23,197,84)(24,198,85)(25,199,86)(26,200,87)(27,201,88)(28,202,89)(29,203,90)(30,204,91)(31,205,92)(32,206,93)(33,207,94)(34,208,95)(35,209,96)(36,210,97)(37,211,98)(38,212,99)(39,213,100)(40,214,101)(41,215,102)(42,216,103)(43,217,104)(44,218,105)(45,219,106)(46,147,107)(47,148,108)(48,149,109)(49,150,110)(50,151,111)(51,152,112)(52,153,113)(53,154,114)(54,155,115)(55,156,116)(56,157,117)(57,158,118)(58,159,119)(59,160,120)(60,161,121)(61,162,122)(62,163,123)(63,164,124)(64,165,125)(65,166,126)(66,167,127)(67,168,128)(68,169,129)(69,170,130)(70,171,131)(71,172,132)(72,173,133)(73,174,134), (74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,181)(142,182)(143,183)(144,184)(145,185)(146,186) );

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),(147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219)], [(1,175,135),(2,176,136),(3,177,137),(4,178,138),(5,179,139),(6,180,140),(7,181,141),(8,182,142),(9,183,143),(10,184,144),(11,185,145),(12,186,146),(13,187,74),(14,188,75),(15,189,76),(16,190,77),(17,191,78),(18,192,79),(19,193,80),(20,194,81),(21,195,82),(22,196,83),(23,197,84),(24,198,85),(25,199,86),(26,200,87),(27,201,88),(28,202,89),(29,203,90),(30,204,91),(31,205,92),(32,206,93),(33,207,94),(34,208,95),(35,209,96),(36,210,97),(37,211,98),(38,212,99),(39,213,100),(40,214,101),(41,215,102),(42,216,103),(43,217,104),(44,218,105),(45,219,106),(46,147,107),(47,148,108),(48,149,109),(49,150,110),(50,151,111),(51,152,112),(52,153,113),(53,154,114),(54,155,115),(55,156,116),(56,157,117),(57,158,118),(58,159,119),(59,160,120),(60,161,121),(61,162,122),(62,163,123),(63,164,124),(64,165,125),(65,166,126),(66,167,127),(67,168,128),(68,169,129),(69,170,130),(70,171,131),(71,172,132),(72,173,133),(73,174,134)], [(74,187),(75,188),(76,189),(77,190),(78,191),(79,192),(80,193),(81,194),(82,195),(83,196),(84,197),(85,198),(86,199),(87,200),(88,201),(89,202),(90,203),(91,204),(92,205),(93,206),(94,207),(95,208),(96,209),(97,210),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,217),(105,218),(106,219),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160),(121,161),(122,162),(123,163),(124,164),(125,165),(126,166),(127,167),(128,168),(129,169),(130,170),(131,171),(132,172),(133,173),(134,174),(135,175),(136,176),(137,177),(138,178),(139,179),(140,180),(141,181),(142,182),(143,183),(144,184),(145,185),(146,186)])

219 conjugacy classes

class 1  2  3 73A···73BT146A···146BT219A···219BT
order12373···73146···146219···219
size1321···13···32···2

219 irreducible representations

dim111122
type+++
imageC1C2C73C146S3S3×C73
kernelS3×C73C219S3C3C73C1
# reps117272172

Matrix representation of S3×C73 in GL2(𝔽439) generated by

4330
0433
,
0438
1438
,
1438
0438
G:=sub<GL(2,GF(439))| [433,0,0,433],[0,1,438,438],[1,0,438,438] >;

S3×C73 in GAP, Magma, Sage, TeX

S_3\times C_{73}
% in TeX

G:=Group("S3xC73");
// GroupNames label

G:=SmallGroup(438,3);
// by ID

G=gap.SmallGroup(438,3);
# by ID

G:=PCGroup([3,-2,-73,-3,2630]);
// Polycyclic

G:=Group<a,b,c|a^73=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×C73 in TeX

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