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G = S3×C79order 474 = 2·3·79

Direct product of C79 and S3

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

Aliases: S3×C79, C3⋊C158, C2373C2, SmallGroup(474,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C79
C1C3C237 — S3×C79
C3 — S3×C79
C1C79

Generators and relations for S3×C79
 G = < a,b,c | a79=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C158

Smallest permutation representation of S3×C79
On 237 points
Generators in S237
(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)(159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237)
(1 105 194)(2 106 195)(3 107 196)(4 108 197)(5 109 198)(6 110 199)(7 111 200)(8 112 201)(9 113 202)(10 114 203)(11 115 204)(12 116 205)(13 117 206)(14 118 207)(15 119 208)(16 120 209)(17 121 210)(18 122 211)(19 123 212)(20 124 213)(21 125 214)(22 126 215)(23 127 216)(24 128 217)(25 129 218)(26 130 219)(27 131 220)(28 132 221)(29 133 222)(30 134 223)(31 135 224)(32 136 225)(33 137 226)(34 138 227)(35 139 228)(36 140 229)(37 141 230)(38 142 231)(39 143 232)(40 144 233)(41 145 234)(42 146 235)(43 147 236)(44 148 237)(45 149 159)(46 150 160)(47 151 161)(48 152 162)(49 153 163)(50 154 164)(51 155 165)(52 156 166)(53 157 167)(54 158 168)(55 80 169)(56 81 170)(57 82 171)(58 83 172)(59 84 173)(60 85 174)(61 86 175)(62 87 176)(63 88 177)(64 89 178)(65 90 179)(66 91 180)(67 92 181)(68 93 182)(69 94 183)(70 95 184)(71 96 185)(72 97 186)(73 98 187)(74 99 188)(75 100 189)(76 101 190)(77 102 191)(78 103 192)(79 104 193)
(80 169)(81 170)(82 171)(83 172)(84 173)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 183)(95 184)(96 185)(97 186)(98 187)(99 188)(100 189)(101 190)(102 191)(103 192)(104 193)(105 194)(106 195)(107 196)(108 197)(109 198)(110 199)(111 200)(112 201)(113 202)(114 203)(115 204)(116 205)(117 206)(118 207)(119 208)(120 209)(121 210)(122 211)(123 212)(124 213)(125 214)(126 215)(127 216)(128 217)(129 218)(130 219)(131 220)(132 221)(133 222)(134 223)(135 224)(136 225)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 233)(145 234)(146 235)(147 236)(148 237)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)(157 167)(158 168)

G:=sub<Sym(237)| (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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,105,194)(2,106,195)(3,107,196)(4,108,197)(5,109,198)(6,110,199)(7,111,200)(8,112,201)(9,113,202)(10,114,203)(11,115,204)(12,116,205)(13,117,206)(14,118,207)(15,119,208)(16,120,209)(17,121,210)(18,122,211)(19,123,212)(20,124,213)(21,125,214)(22,126,215)(23,127,216)(24,128,217)(25,129,218)(26,130,219)(27,131,220)(28,132,221)(29,133,222)(30,134,223)(31,135,224)(32,136,225)(33,137,226)(34,138,227)(35,139,228)(36,140,229)(37,141,230)(38,142,231)(39,143,232)(40,144,233)(41,145,234)(42,146,235)(43,147,236)(44,148,237)(45,149,159)(46,150,160)(47,151,161)(48,152,162)(49,153,163)(50,154,164)(51,155,165)(52,156,166)(53,157,167)(54,158,168)(55,80,169)(56,81,170)(57,82,171)(58,83,172)(59,84,173)(60,85,174)(61,86,175)(62,87,176)(63,88,177)(64,89,178)(65,90,179)(66,91,180)(67,92,181)(68,93,182)(69,94,183)(70,95,184)(71,96,185)(72,97,186)(73,98,187)(74,99,188)(75,100,189)(76,101,190)(77,102,191)(78,103,192)(79,104,193), (80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(109,198)(110,199)(111,200)(112,201)(113,202)(114,203)(115,204)(116,205)(117,206)(118,207)(119,208)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,225)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(157,167)(158,168)>;

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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,105,194)(2,106,195)(3,107,196)(4,108,197)(5,109,198)(6,110,199)(7,111,200)(8,112,201)(9,113,202)(10,114,203)(11,115,204)(12,116,205)(13,117,206)(14,118,207)(15,119,208)(16,120,209)(17,121,210)(18,122,211)(19,123,212)(20,124,213)(21,125,214)(22,126,215)(23,127,216)(24,128,217)(25,129,218)(26,130,219)(27,131,220)(28,132,221)(29,133,222)(30,134,223)(31,135,224)(32,136,225)(33,137,226)(34,138,227)(35,139,228)(36,140,229)(37,141,230)(38,142,231)(39,143,232)(40,144,233)(41,145,234)(42,146,235)(43,147,236)(44,148,237)(45,149,159)(46,150,160)(47,151,161)(48,152,162)(49,153,163)(50,154,164)(51,155,165)(52,156,166)(53,157,167)(54,158,168)(55,80,169)(56,81,170)(57,82,171)(58,83,172)(59,84,173)(60,85,174)(61,86,175)(62,87,176)(63,88,177)(64,89,178)(65,90,179)(66,91,180)(67,92,181)(68,93,182)(69,94,183)(70,95,184)(71,96,185)(72,97,186)(73,98,187)(74,99,188)(75,100,189)(76,101,190)(77,102,191)(78,103,192)(79,104,193), (80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(109,198)(110,199)(111,200)(112,201)(113,202)(114,203)(115,204)(116,205)(117,206)(118,207)(119,208)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,225)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)(157,167)(158,168) );

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),(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237)], [(1,105,194),(2,106,195),(3,107,196),(4,108,197),(5,109,198),(6,110,199),(7,111,200),(8,112,201),(9,113,202),(10,114,203),(11,115,204),(12,116,205),(13,117,206),(14,118,207),(15,119,208),(16,120,209),(17,121,210),(18,122,211),(19,123,212),(20,124,213),(21,125,214),(22,126,215),(23,127,216),(24,128,217),(25,129,218),(26,130,219),(27,131,220),(28,132,221),(29,133,222),(30,134,223),(31,135,224),(32,136,225),(33,137,226),(34,138,227),(35,139,228),(36,140,229),(37,141,230),(38,142,231),(39,143,232),(40,144,233),(41,145,234),(42,146,235),(43,147,236),(44,148,237),(45,149,159),(46,150,160),(47,151,161),(48,152,162),(49,153,163),(50,154,164),(51,155,165),(52,156,166),(53,157,167),(54,158,168),(55,80,169),(56,81,170),(57,82,171),(58,83,172),(59,84,173),(60,85,174),(61,86,175),(62,87,176),(63,88,177),(64,89,178),(65,90,179),(66,91,180),(67,92,181),(68,93,182),(69,94,183),(70,95,184),(71,96,185),(72,97,186),(73,98,187),(74,99,188),(75,100,189),(76,101,190),(77,102,191),(78,103,192),(79,104,193)], [(80,169),(81,170),(82,171),(83,172),(84,173),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,181),(93,182),(94,183),(95,184),(96,185),(97,186),(98,187),(99,188),(100,189),(101,190),(102,191),(103,192),(104,193),(105,194),(106,195),(107,196),(108,197),(109,198),(110,199),(111,200),(112,201),(113,202),(114,203),(115,204),(116,205),(117,206),(118,207),(119,208),(120,209),(121,210),(122,211),(123,212),(124,213),(125,214),(126,215),(127,216),(128,217),(129,218),(130,219),(131,220),(132,221),(133,222),(134,223),(135,224),(136,225),(137,226),(138,227),(139,228),(140,229),(141,230),(142,231),(143,232),(144,233),(145,234),(146,235),(147,236),(148,237),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(155,165),(156,166),(157,167),(158,168)])

237 conjugacy classes

class 1  2  3 79A···79BZ158A···158BZ237A···237BZ
order12379···79158···158237···237
size1321···13···32···2

237 irreducible representations

dim111122
type+++
imageC1C2C79C158S3S3×C79
kernelS3×C79C237S3C3C79C1
# reps117878178

Matrix representation of S3×C79 in GL2(𝔽1423) generated by

8360
0836
,
14221422
10
,
10
14221422
G:=sub<GL(2,GF(1423))| [836,0,0,836],[1422,1,1422,0],[1,1422,0,1422] >;

S3×C79 in GAP, Magma, Sage, TeX

S_3\times C_{79}
% in TeX

G:=Group("S3xC79");
// GroupNames label

G:=SmallGroup(474,3);
// by ID

G=gap.SmallGroup(474,3);
# by ID

G:=PCGroup([3,-2,-79,-3,2846]);
// Polycyclic

G:=Group<a,b,c|a^79=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×C79 in TeX

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