direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D37, C74⋊3C6, C222⋊2C2, C111⋊3C22, C37⋊3(C2×C6), SmallGroup(444,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C6×D37 |
Generators and relations for C6×D37
G = < a,b,c | a6=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D37
►On 222 points
Generators in S222
(1 162 104 147 39 221)(2 163 105 148 40 222)(3 164 106 112 41 186)(4 165 107 113 42 187)(5 166 108 114 43 188)(6 167 109 115 44 189)(7 168 110 116 45 190)(8 169 111 117 46 191)(9 170 75 118 47 192)(10 171 76 119 48 193)(11 172 77 120 49 194)(12 173 78 121 50 195)(13 174 79 122 51 196)(14 175 80 123 52 197)(15 176 81 124 53 198)(16 177 82 125 54 199)(17 178 83 126 55 200)(18 179 84 127 56 201)(19 180 85 128 57 202)(20 181 86 129 58 203)(21 182 87 130 59 204)(22 183 88 131 60 205)(23 184 89 132 61 206)(24 185 90 133 62 207)(25 149 91 134 63 208)(26 150 92 135 64 209)(27 151 93 136 65 210)(28 152 94 137 66 211)(29 153 95 138 67 212)(30 154 96 139 68 213)(31 155 97 140 69 214)(32 156 98 141 70 215)(33 157 99 142 71 216)(34 158 100 143 72 217)(35 159 101 144 73 218)(36 160 102 145 74 219)(37 161 103 146 38 220)
(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)
(1 146)(2 145)(3 144)(4 143)(5 142)(6 141)(7 140)(8 139)(9 138)(10 137)(11 136)(12 135)(13 134)(14 133)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 148)(37 147)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 185)(53 184)(54 183)(55 182)(56 181)(57 180)(58 179)(59 178)(60 177)(61 176)(62 175)(63 174)(64 173)(65 172)(66 171)(67 170)(68 169)(69 168)(70 167)(71 166)(72 165)(73 164)(74 163)(75 212)(76 211)(77 210)(78 209)(79 208)(80 207)(81 206)(82 205)(83 204)(84 203)(85 202)(86 201)(87 200)(88 199)(89 198)(90 197)(91 196)(92 195)(93 194)(94 193)(95 192)(96 191)(97 190)(98 189)(99 188)(100 187)(101 186)(102 222)(103 221)(104 220)(105 219)(106 218)(107 217)(108 216)(109 215)(110 214)(111 213)
G:=sub<Sym(222)| (1,162,104,147,39,221)(2,163,105,148,40,222)(3,164,106,112,41,186)(4,165,107,113,42,187)(5,166,108,114,43,188)(6,167,109,115,44,189)(7,168,110,116,45,190)(8,169,111,117,46,191)(9,170,75,118,47,192)(10,171,76,119,48,193)(11,172,77,120,49,194)(12,173,78,121,50,195)(13,174,79,122,51,196)(14,175,80,123,52,197)(15,176,81,124,53,198)(16,177,82,125,54,199)(17,178,83,126,55,200)(18,179,84,127,56,201)(19,180,85,128,57,202)(20,181,86,129,58,203)(21,182,87,130,59,204)(22,183,88,131,60,205)(23,184,89,132,61,206)(24,185,90,133,62,207)(25,149,91,134,63,208)(26,150,92,135,64,209)(27,151,93,136,65,210)(28,152,94,137,66,211)(29,153,95,138,67,212)(30,154,96,139,68,213)(31,155,97,140,69,214)(32,156,98,141,70,215)(33,157,99,142,71,216)(34,158,100,143,72,217)(35,159,101,144,73,218)(36,160,102,145,74,219)(37,161,103,146,38,220), (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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,148)(37,147)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,185)(53,184)(54,183)(55,182)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,212)(76,211)(77,210)(78,209)(79,208)(80,207)(81,206)(82,205)(83,204)(84,203)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,196)(92,195)(93,194)(94,193)(95,192)(96,191)(97,190)(98,189)(99,188)(100,187)(101,186)(102,222)(103,221)(104,220)(105,219)(106,218)(107,217)(108,216)(109,215)(110,214)(111,213)>;
G:=Group( (1,162,104,147,39,221)(2,163,105,148,40,222)(3,164,106,112,41,186)(4,165,107,113,42,187)(5,166,108,114,43,188)(6,167,109,115,44,189)(7,168,110,116,45,190)(8,169,111,117,46,191)(9,170,75,118,47,192)(10,171,76,119,48,193)(11,172,77,120,49,194)(12,173,78,121,50,195)(13,174,79,122,51,196)(14,175,80,123,52,197)(15,176,81,124,53,198)(16,177,82,125,54,199)(17,178,83,126,55,200)(18,179,84,127,56,201)(19,180,85,128,57,202)(20,181,86,129,58,203)(21,182,87,130,59,204)(22,183,88,131,60,205)(23,184,89,132,61,206)(24,185,90,133,62,207)(25,149,91,134,63,208)(26,150,92,135,64,209)(27,151,93,136,65,210)(28,152,94,137,66,211)(29,153,95,138,67,212)(30,154,96,139,68,213)(31,155,97,140,69,214)(32,156,98,141,70,215)(33,157,99,142,71,216)(34,158,100,143,72,217)(35,159,101,144,73,218)(36,160,102,145,74,219)(37,161,103,146,38,220), (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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,148)(37,147)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,185)(53,184)(54,183)(55,182)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,212)(76,211)(77,210)(78,209)(79,208)(80,207)(81,206)(82,205)(83,204)(84,203)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,196)(92,195)(93,194)(94,193)(95,192)(96,191)(97,190)(98,189)(99,188)(100,187)(101,186)(102,222)(103,221)(104,220)(105,219)(106,218)(107,217)(108,216)(109,215)(110,214)(111,213) );
G=PermutationGroup([[(1,162,104,147,39,221),(2,163,105,148,40,222),(3,164,106,112,41,186),(4,165,107,113,42,187),(5,166,108,114,43,188),(6,167,109,115,44,189),(7,168,110,116,45,190),(8,169,111,117,46,191),(9,170,75,118,47,192),(10,171,76,119,48,193),(11,172,77,120,49,194),(12,173,78,121,50,195),(13,174,79,122,51,196),(14,175,80,123,52,197),(15,176,81,124,53,198),(16,177,82,125,54,199),(17,178,83,126,55,200),(18,179,84,127,56,201),(19,180,85,128,57,202),(20,181,86,129,58,203),(21,182,87,130,59,204),(22,183,88,131,60,205),(23,184,89,132,61,206),(24,185,90,133,62,207),(25,149,91,134,63,208),(26,150,92,135,64,209),(27,151,93,136,65,210),(28,152,94,137,66,211),(29,153,95,138,67,212),(30,154,96,139,68,213),(31,155,97,140,69,214),(32,156,98,141,70,215),(33,157,99,142,71,216),(34,158,100,143,72,217),(35,159,101,144,73,218),(36,160,102,145,74,219),(37,161,103,146,38,220)], [(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)], [(1,146),(2,145),(3,144),(4,143),(5,142),(6,141),(7,140),(8,139),(9,138),(10,137),(11,136),(12,135),(13,134),(14,133),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,126),(22,125),(23,124),(24,123),(25,122),(26,121),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,148),(37,147),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,185),(53,184),(54,183),(55,182),(56,181),(57,180),(58,179),(59,178),(60,177),(61,176),(62,175),(63,174),(64,173),(65,172),(66,171),(67,170),(68,169),(69,168),(70,167),(71,166),(72,165),(73,164),(74,163),(75,212),(76,211),(77,210),(78,209),(79,208),(80,207),(81,206),(82,205),(83,204),(84,203),(85,202),(86,201),(87,200),(88,199),(89,198),(90,197),(91,196),(92,195),(93,194),(94,193),(95,192),(96,191),(97,190),(98,189),(99,188),(100,187),(101,186),(102,222),(103,221),(104,220),(105,219),(106,218),(107,217),(108,216),(109,215),(110,214),(111,213)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 37A | ··· | 37R | 74A | ··· | 74R | 111A | ··· | 111AJ | 222A | ··· | 222AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 | 222 | ··· | 222 |
| size | 1 | 1 | 37 | 37 | 1 | 1 | 1 | 1 | 37 | 37 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D37 | D74 | C3×D37 | C6×D37 |
| kernel | C6×D37 | C3×D37 | C222 | D74 | D37 | C74 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 18 | 18 | 36 | 36 |
Matrix representation of C6×D37 ►in GL3(𝔽223) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 115 | 1 |
| 0 | 59 | 49 |
| 1 | 0 | 0 |
| 0 | 216 | 164 |
| 0 | 152 | 7 |
G:=sub<GL(3,GF(223))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,115,59,0,1,49],[1,0,0,0,216,152,0,164,7] >;
C6×D37 in GAP, Magma, Sage, TeX
C_6\times D_{37} % in TeX
G:=Group("C6xD37"); // GroupNames label
G:=SmallGroup(444,15);
// by ID
G=gap.SmallGroup(444,15);
# by ID
G:=PCGroup([4,-2,-2,-3,-37,6915]);
// Polycyclic
G:=Group<a,b,c|a^6=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export