direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C19×D12, C57⋊6D4, C76⋊3S3, C12⋊1C38, C228⋊5C2, D6⋊1C38, C38.15D6, C114.20C22, C4⋊(S3×C19), C3⋊1(D4×C19), (S3×C38)⋊4C2, C2.4(S3×C38), C6.3(C2×C38), SmallGroup(456,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C19×D12
G = < a,b,c | a19=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C19×D12
►On 228 points
Generators in S228
(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)
(1 146 69 164 110 88 25 195 180 128 221 57)(2 147 70 165 111 89 26 196 181 129 222 39)(3 148 71 166 112 90 27 197 182 130 223 40)(4 149 72 167 113 91 28 198 183 131 224 41)(5 150 73 168 114 92 29 199 184 132 225 42)(6 151 74 169 96 93 30 200 185 133 226 43)(7 152 75 170 97 94 31 201 186 115 227 44)(8 134 76 171 98 95 32 202 187 116 228 45)(9 135 58 153 99 77 33 203 188 117 210 46)(10 136 59 154 100 78 34 204 189 118 211 47)(11 137 60 155 101 79 35 205 190 119 212 48)(12 138 61 156 102 80 36 206 172 120 213 49)(13 139 62 157 103 81 37 207 173 121 214 50)(14 140 63 158 104 82 38 208 174 122 215 51)(15 141 64 159 105 83 20 209 175 123 216 52)(16 142 65 160 106 84 21 191 176 124 217 53)(17 143 66 161 107 85 22 192 177 125 218 54)(18 144 67 162 108 86 23 193 178 126 219 55)(19 145 68 163 109 87 24 194 179 127 220 56)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 175)(21 176)(22 177)(23 178)(24 179)(25 180)(26 181)(27 182)(28 183)(29 184)(30 185)(31 186)(32 187)(33 188)(34 189)(35 190)(36 172)(37 173)(38 174)(39 165)(40 166)(41 167)(42 168)(43 169)(44 170)(45 171)(46 153)(47 154)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 164)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 115)(95 116)(96 226)(97 227)(98 228)(99 210)(100 211)(101 212)(102 213)(103 214)(104 215)(105 216)(106 217)(107 218)(108 219)(109 220)(110 221)(111 222)(112 223)(113 224)(114 225)
G:=sub<Sym(228)| (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), (1,146,69,164,110,88,25,195,180,128,221,57)(2,147,70,165,111,89,26,196,181,129,222,39)(3,148,71,166,112,90,27,197,182,130,223,40)(4,149,72,167,113,91,28,198,183,131,224,41)(5,150,73,168,114,92,29,199,184,132,225,42)(6,151,74,169,96,93,30,200,185,133,226,43)(7,152,75,170,97,94,31,201,186,115,227,44)(8,134,76,171,98,95,32,202,187,116,228,45)(9,135,58,153,99,77,33,203,188,117,210,46)(10,136,59,154,100,78,34,204,189,118,211,47)(11,137,60,155,101,79,35,205,190,119,212,48)(12,138,61,156,102,80,36,206,172,120,213,49)(13,139,62,157,103,81,37,207,173,121,214,50)(14,140,63,158,104,82,38,208,174,122,215,51)(15,141,64,159,105,83,20,209,175,123,216,52)(16,142,65,160,106,84,21,191,176,124,217,53)(17,143,66,161,107,85,22,192,177,125,218,54)(18,144,67,162,108,86,23,193,178,126,219,55)(19,145,68,163,109,87,24,194,179,127,220,56), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,175)(21,176)(22,177)(23,178)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,185)(31,186)(32,187)(33,188)(34,189)(35,190)(36,172)(37,173)(38,174)(39,165)(40,166)(41,167)(42,168)(43,169)(44,170)(45,171)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,115)(95,116)(96,226)(97,227)(98,228)(99,210)(100,211)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,221)(111,222)(112,223)(113,224)(114,225)>;
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), (1,146,69,164,110,88,25,195,180,128,221,57)(2,147,70,165,111,89,26,196,181,129,222,39)(3,148,71,166,112,90,27,197,182,130,223,40)(4,149,72,167,113,91,28,198,183,131,224,41)(5,150,73,168,114,92,29,199,184,132,225,42)(6,151,74,169,96,93,30,200,185,133,226,43)(7,152,75,170,97,94,31,201,186,115,227,44)(8,134,76,171,98,95,32,202,187,116,228,45)(9,135,58,153,99,77,33,203,188,117,210,46)(10,136,59,154,100,78,34,204,189,118,211,47)(11,137,60,155,101,79,35,205,190,119,212,48)(12,138,61,156,102,80,36,206,172,120,213,49)(13,139,62,157,103,81,37,207,173,121,214,50)(14,140,63,158,104,82,38,208,174,122,215,51)(15,141,64,159,105,83,20,209,175,123,216,52)(16,142,65,160,106,84,21,191,176,124,217,53)(17,143,66,161,107,85,22,192,177,125,218,54)(18,144,67,162,108,86,23,193,178,126,219,55)(19,145,68,163,109,87,24,194,179,127,220,56), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,175)(21,176)(22,177)(23,178)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,185)(31,186)(32,187)(33,188)(34,189)(35,190)(36,172)(37,173)(38,174)(39,165)(40,166)(41,167)(42,168)(43,169)(44,170)(45,171)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,115)(95,116)(96,226)(97,227)(98,228)(99,210)(100,211)(101,212)(102,213)(103,214)(104,215)(105,216)(106,217)(107,218)(108,219)(109,220)(110,221)(111,222)(112,223)(113,224)(114,225) );
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)], [(1,146,69,164,110,88,25,195,180,128,221,57),(2,147,70,165,111,89,26,196,181,129,222,39),(3,148,71,166,112,90,27,197,182,130,223,40),(4,149,72,167,113,91,28,198,183,131,224,41),(5,150,73,168,114,92,29,199,184,132,225,42),(6,151,74,169,96,93,30,200,185,133,226,43),(7,152,75,170,97,94,31,201,186,115,227,44),(8,134,76,171,98,95,32,202,187,116,228,45),(9,135,58,153,99,77,33,203,188,117,210,46),(10,136,59,154,100,78,34,204,189,118,211,47),(11,137,60,155,101,79,35,205,190,119,212,48),(12,138,61,156,102,80,36,206,172,120,213,49),(13,139,62,157,103,81,37,207,173,121,214,50),(14,140,63,158,104,82,38,208,174,122,215,51),(15,141,64,159,105,83,20,209,175,123,216,52),(16,142,65,160,106,84,21,191,176,124,217,53),(17,143,66,161,107,85,22,192,177,125,218,54),(18,144,67,162,108,86,23,193,178,126,219,55),(19,145,68,163,109,87,24,194,179,127,220,56)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,175),(21,176),(22,177),(23,178),(24,179),(25,180),(26,181),(27,182),(28,183),(29,184),(30,185),(31,186),(32,187),(33,188),(34,189),(35,190),(36,172),(37,173),(38,174),(39,165),(40,166),(41,167),(42,168),(43,169),(44,170),(45,171),(46,153),(47,154),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,164),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,115),(95,116),(96,226),(97,227),(98,228),(99,210),(100,211),(101,212),(102,213),(103,214),(104,215),(105,216),(106,217),(107,218),(108,219),(109,220),(110,221),(111,222),(112,223),(113,224),(114,225)]])
171 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BB | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
171 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | S3 | D4 | D6 | D12 | S3×C19 | D4×C19 | S3×C38 | C19×D12 |
| kernel | C19×D12 | C228 | S3×C38 | D12 | C12 | D6 | C76 | C57 | C38 | C19 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 1 | 1 | 1 | 2 | 18 | 18 | 18 | 36 |
Matrix representation of C19×D12 ►in GL4(𝔽229) generated by
| 225 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 44 |
| 174 | 223 | 0 | 0 |
| 199 | 55 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 228 | 0 |
| 228 | 0 | 0 | 0 |
| 171 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 228 | 228 |
G:=sub<GL(4,GF(229))| [225,0,0,0,0,225,0,0,0,0,44,0,0,0,0,44],[174,199,0,0,223,55,0,0,0,0,1,228,0,0,1,0],[228,171,0,0,0,1,0,0,0,0,1,228,0,0,0,228] >;
C19×D12 in GAP, Magma, Sage, TeX
C_{19}\times D_{12} % in TeX
G:=Group("C19xD12"); // GroupNames label
G:=SmallGroup(456,31);
// by ID
G=gap.SmallGroup(456,31);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-3,781,386,7604]);
// Polycyclic
G:=Group<a,b,c|a^19=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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