direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C74, C6⋊C74, C222⋊3C2, C111⋊4C22, C3⋊(C2×C74), SmallGroup(444,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C74 |
Generators and relations for S3×C74
G = < a,b,c | a74=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C74
►On 222 points
Generators in S222
(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)
(1 84 210)(2 85 211)(3 86 212)(4 87 213)(5 88 214)(6 89 215)(7 90 216)(8 91 217)(9 92 218)(10 93 219)(11 94 220)(12 95 221)(13 96 222)(14 97 149)(15 98 150)(16 99 151)(17 100 152)(18 101 153)(19 102 154)(20 103 155)(21 104 156)(22 105 157)(23 106 158)(24 107 159)(25 108 160)(26 109 161)(27 110 162)(28 111 163)(29 112 164)(30 113 165)(31 114 166)(32 115 167)(33 116 168)(34 117 169)(35 118 170)(36 119 171)(37 120 172)(38 121 173)(39 122 174)(40 123 175)(41 124 176)(42 125 177)(43 126 178)(44 127 179)(45 128 180)(46 129 181)(47 130 182)(48 131 183)(49 132 184)(50 133 185)(51 134 186)(52 135 187)(53 136 188)(54 137 189)(55 138 190)(56 139 191)(57 140 192)(58 141 193)(59 142 194)(60 143 195)(61 144 196)(62 145 197)(63 146 198)(64 147 199)(65 148 200)(66 75 201)(67 76 202)(68 77 203)(69 78 204)(70 79 205)(71 80 206)(72 81 207)(73 82 208)(74 83 209)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(75 164)(76 165)(77 166)(78 167)(79 168)(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)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)
G:=sub<Sym(222)| (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), (1,84,210)(2,85,211)(3,86,212)(4,87,213)(5,88,214)(6,89,215)(7,90,216)(8,91,217)(9,92,218)(10,93,219)(11,94,220)(12,95,221)(13,96,222)(14,97,149)(15,98,150)(16,99,151)(17,100,152)(18,101,153)(19,102,154)(20,103,155)(21,104,156)(22,105,157)(23,106,158)(24,107,159)(25,108,160)(26,109,161)(27,110,162)(28,111,163)(29,112,164)(30,113,165)(31,114,166)(32,115,167)(33,116,168)(34,117,169)(35,118,170)(36,119,171)(37,120,172)(38,121,173)(39,122,174)(40,123,175)(41,124,176)(42,125,177)(43,126,178)(44,127,179)(45,128,180)(46,129,181)(47,130,182)(48,131,183)(49,132,184)(50,133,185)(51,134,186)(52,135,187)(53,136,188)(54,137,189)(55,138,190)(56,139,191)(57,140,192)(58,141,193)(59,142,194)(60,143,195)(61,144,196)(62,145,197)(63,146,198)(64,147,199)(65,148,200)(66,75,201)(67,76,202)(68,77,203)(69,78,204)(70,79,205)(71,80,206)(72,81,207)(73,82,208)(74,83,209), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(75,164)(76,165)(77,166)(78,167)(79,168)(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)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)>;
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), (1,84,210)(2,85,211)(3,86,212)(4,87,213)(5,88,214)(6,89,215)(7,90,216)(8,91,217)(9,92,218)(10,93,219)(11,94,220)(12,95,221)(13,96,222)(14,97,149)(15,98,150)(16,99,151)(17,100,152)(18,101,153)(19,102,154)(20,103,155)(21,104,156)(22,105,157)(23,106,158)(24,107,159)(25,108,160)(26,109,161)(27,110,162)(28,111,163)(29,112,164)(30,113,165)(31,114,166)(32,115,167)(33,116,168)(34,117,169)(35,118,170)(36,119,171)(37,120,172)(38,121,173)(39,122,174)(40,123,175)(41,124,176)(42,125,177)(43,126,178)(44,127,179)(45,128,180)(46,129,181)(47,130,182)(48,131,183)(49,132,184)(50,133,185)(51,134,186)(52,135,187)(53,136,188)(54,137,189)(55,138,190)(56,139,191)(57,140,192)(58,141,193)(59,142,194)(60,143,195)(61,144,196)(62,145,197)(63,146,198)(64,147,199)(65,148,200)(66,75,201)(67,76,202)(68,77,203)(69,78,204)(70,79,205)(71,80,206)(72,81,207)(73,82,208)(74,83,209), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(75,164)(76,165)(77,166)(78,167)(79,168)(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)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163) );
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)], [(1,84,210),(2,85,211),(3,86,212),(4,87,213),(5,88,214),(6,89,215),(7,90,216),(8,91,217),(9,92,218),(10,93,219),(11,94,220),(12,95,221),(13,96,222),(14,97,149),(15,98,150),(16,99,151),(17,100,152),(18,101,153),(19,102,154),(20,103,155),(21,104,156),(22,105,157),(23,106,158),(24,107,159),(25,108,160),(26,109,161),(27,110,162),(28,111,163),(29,112,164),(30,113,165),(31,114,166),(32,115,167),(33,116,168),(34,117,169),(35,118,170),(36,119,171),(37,120,172),(38,121,173),(39,122,174),(40,123,175),(41,124,176),(42,125,177),(43,126,178),(44,127,179),(45,128,180),(46,129,181),(47,130,182),(48,131,183),(49,132,184),(50,133,185),(51,134,186),(52,135,187),(53,136,188),(54,137,189),(55,138,190),(56,139,191),(57,140,192),(58,141,193),(59,142,194),(60,143,195),(61,144,196),(62,145,197),(63,146,198),(64,147,199),(65,148,200),(66,75,201),(67,76,202),(68,77,203),(69,78,204),(70,79,205),(71,80,206),(72,81,207),(73,82,208),(74,83,209)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(75,164),(76,165),(77,166),(78,167),(79,168),(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),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,162),(148,163)]])
222 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 74AK | ··· | 74DD | 111A | ··· | 111AJ | 222A | ··· | 222AJ |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 37 | ··· | 37 | 74 | ··· | 74 | 74 | ··· | 74 | 111 | ··· | 111 | 222 | ··· | 222 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
222 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C37 | C74 | C74 | S3 | D6 | S3×C37 | S3×C74 |
| kernel | S3×C74 | S3×C37 | C222 | D6 | S3 | C6 | C74 | C37 | C2 | C1 |
| # reps | 1 | 2 | 1 | 36 | 72 | 36 | 1 | 1 | 36 | 36 |
Matrix representation of S3×C74 ►in GL3(𝔽223) generated by
| 222 | 0 | 0 |
| 0 | 68 | 0 |
| 0 | 0 | 68 |
| 1 | 0 | 0 |
| 0 | 222 | 222 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 222 | 222 |
G:=sub<GL(3,GF(223))| [222,0,0,0,68,0,0,0,68],[1,0,0,0,222,1,0,222,0],[1,0,0,0,1,222,0,0,222] >;
S3×C74 in GAP, Magma, Sage, TeX
S_3\times C_{74} % in TeX
G:=Group("S3xC74"); // GroupNames label
G:=SmallGroup(444,16);
// by ID
G=gap.SmallGroup(444,16);
# by ID
G:=PCGroup([4,-2,-2,-37,-3,4739]);
// Polycyclic
G:=Group<a,b,c|a^74=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export