direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic19, C57⋊2C4, C19⋊3C12, C38.3C6, C6.2D19, C114.2C2, C2.(C3×D19), SmallGroup(228,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C3×Dic19 |
Generators and relations for C3×Dic19
G = < a,b,c | a3=b38=1, c2=b19, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic19
►Regular action on 228 points
Generators in S228
(1 106 67)(2 107 68)(3 108 69)(4 109 70)(5 110 71)(6 111 72)(7 112 73)(8 113 74)(9 114 75)(10 77 76)(11 78 39)(12 79 40)(13 80 41)(14 81 42)(15 82 43)(16 83 44)(17 84 45)(18 85 46)(19 86 47)(20 87 48)(21 88 49)(22 89 50)(23 90 51)(24 91 52)(25 92 53)(26 93 54)(27 94 55)(28 95 56)(29 96 57)(30 97 58)(31 98 59)(32 99 60)(33 100 61)(34 101 62)(35 102 63)(36 103 64)(37 104 65)(38 105 66)(115 210 153)(116 211 154)(117 212 155)(118 213 156)(119 214 157)(120 215 158)(121 216 159)(122 217 160)(123 218 161)(124 219 162)(125 220 163)(126 221 164)(127 222 165)(128 223 166)(129 224 167)(130 225 168)(131 226 169)(132 227 170)(133 228 171)(134 191 172)(135 192 173)(136 193 174)(137 194 175)(138 195 176)(139 196 177)(140 197 178)(141 198 179)(142 199 180)(143 200 181)(144 201 182)(145 202 183)(146 203 184)(147 204 185)(148 205 186)(149 206 187)(150 207 188)(151 208 189)(152 209 190)
(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)
(1 115 20 134)(2 152 21 133)(3 151 22 132)(4 150 23 131)(5 149 24 130)(6 148 25 129)(7 147 26 128)(8 146 27 127)(9 145 28 126)(10 144 29 125)(11 143 30 124)(12 142 31 123)(13 141 32 122)(14 140 33 121)(15 139 34 120)(16 138 35 119)(17 137 36 118)(18 136 37 117)(19 135 38 116)(39 181 58 162)(40 180 59 161)(41 179 60 160)(42 178 61 159)(43 177 62 158)(44 176 63 157)(45 175 64 156)(46 174 65 155)(47 173 66 154)(48 172 67 153)(49 171 68 190)(50 170 69 189)(51 169 70 188)(52 168 71 187)(53 167 72 186)(54 166 73 185)(55 165 74 184)(56 164 75 183)(57 163 76 182)(77 201 96 220)(78 200 97 219)(79 199 98 218)(80 198 99 217)(81 197 100 216)(82 196 101 215)(83 195 102 214)(84 194 103 213)(85 193 104 212)(86 192 105 211)(87 191 106 210)(88 228 107 209)(89 227 108 208)(90 226 109 207)(91 225 110 206)(92 224 111 205)(93 223 112 204)(94 222 113 203)(95 221 114 202)
G:=sub<Sym(228)| (1,106,67)(2,107,68)(3,108,69)(4,109,70)(5,110,71)(6,111,72)(7,112,73)(8,113,74)(9,114,75)(10,77,76)(11,78,39)(12,79,40)(13,80,41)(14,81,42)(15,82,43)(16,83,44)(17,84,45)(18,85,46)(19,86,47)(20,87,48)(21,88,49)(22,89,50)(23,90,51)(24,91,52)(25,92,53)(26,93,54)(27,94,55)(28,95,56)(29,96,57)(30,97,58)(31,98,59)(32,99,60)(33,100,61)(34,101,62)(35,102,63)(36,103,64)(37,104,65)(38,105,66)(115,210,153)(116,211,154)(117,212,155)(118,213,156)(119,214,157)(120,215,158)(121,216,159)(122,217,160)(123,218,161)(124,219,162)(125,220,163)(126,221,164)(127,222,165)(128,223,166)(129,224,167)(130,225,168)(131,226,169)(132,227,170)(133,228,171)(134,191,172)(135,192,173)(136,193,174)(137,194,175)(138,195,176)(139,196,177)(140,197,178)(141,198,179)(142,199,180)(143,200,181)(144,201,182)(145,202,183)(146,203,184)(147,204,185)(148,205,186)(149,206,187)(150,207,188)(151,208,189)(152,209,190), (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), (1,115,20,134)(2,152,21,133)(3,151,22,132)(4,150,23,131)(5,149,24,130)(6,148,25,129)(7,147,26,128)(8,146,27,127)(9,145,28,126)(10,144,29,125)(11,143,30,124)(12,142,31,123)(13,141,32,122)(14,140,33,121)(15,139,34,120)(16,138,35,119)(17,137,36,118)(18,136,37,117)(19,135,38,116)(39,181,58,162)(40,180,59,161)(41,179,60,160)(42,178,61,159)(43,177,62,158)(44,176,63,157)(45,175,64,156)(46,174,65,155)(47,173,66,154)(48,172,67,153)(49,171,68,190)(50,170,69,189)(51,169,70,188)(52,168,71,187)(53,167,72,186)(54,166,73,185)(55,165,74,184)(56,164,75,183)(57,163,76,182)(77,201,96,220)(78,200,97,219)(79,199,98,218)(80,198,99,217)(81,197,100,216)(82,196,101,215)(83,195,102,214)(84,194,103,213)(85,193,104,212)(86,192,105,211)(87,191,106,210)(88,228,107,209)(89,227,108,208)(90,226,109,207)(91,225,110,206)(92,224,111,205)(93,223,112,204)(94,222,113,203)(95,221,114,202)>;
G:=Group( (1,106,67)(2,107,68)(3,108,69)(4,109,70)(5,110,71)(6,111,72)(7,112,73)(8,113,74)(9,114,75)(10,77,76)(11,78,39)(12,79,40)(13,80,41)(14,81,42)(15,82,43)(16,83,44)(17,84,45)(18,85,46)(19,86,47)(20,87,48)(21,88,49)(22,89,50)(23,90,51)(24,91,52)(25,92,53)(26,93,54)(27,94,55)(28,95,56)(29,96,57)(30,97,58)(31,98,59)(32,99,60)(33,100,61)(34,101,62)(35,102,63)(36,103,64)(37,104,65)(38,105,66)(115,210,153)(116,211,154)(117,212,155)(118,213,156)(119,214,157)(120,215,158)(121,216,159)(122,217,160)(123,218,161)(124,219,162)(125,220,163)(126,221,164)(127,222,165)(128,223,166)(129,224,167)(130,225,168)(131,226,169)(132,227,170)(133,228,171)(134,191,172)(135,192,173)(136,193,174)(137,194,175)(138,195,176)(139,196,177)(140,197,178)(141,198,179)(142,199,180)(143,200,181)(144,201,182)(145,202,183)(146,203,184)(147,204,185)(148,205,186)(149,206,187)(150,207,188)(151,208,189)(152,209,190), (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), (1,115,20,134)(2,152,21,133)(3,151,22,132)(4,150,23,131)(5,149,24,130)(6,148,25,129)(7,147,26,128)(8,146,27,127)(9,145,28,126)(10,144,29,125)(11,143,30,124)(12,142,31,123)(13,141,32,122)(14,140,33,121)(15,139,34,120)(16,138,35,119)(17,137,36,118)(18,136,37,117)(19,135,38,116)(39,181,58,162)(40,180,59,161)(41,179,60,160)(42,178,61,159)(43,177,62,158)(44,176,63,157)(45,175,64,156)(46,174,65,155)(47,173,66,154)(48,172,67,153)(49,171,68,190)(50,170,69,189)(51,169,70,188)(52,168,71,187)(53,167,72,186)(54,166,73,185)(55,165,74,184)(56,164,75,183)(57,163,76,182)(77,201,96,220)(78,200,97,219)(79,199,98,218)(80,198,99,217)(81,197,100,216)(82,196,101,215)(83,195,102,214)(84,194,103,213)(85,193,104,212)(86,192,105,211)(87,191,106,210)(88,228,107,209)(89,227,108,208)(90,226,109,207)(91,225,110,206)(92,224,111,205)(93,223,112,204)(94,222,113,203)(95,221,114,202) );
G=PermutationGroup([[(1,106,67),(2,107,68),(3,108,69),(4,109,70),(5,110,71),(6,111,72),(7,112,73),(8,113,74),(9,114,75),(10,77,76),(11,78,39),(12,79,40),(13,80,41),(14,81,42),(15,82,43),(16,83,44),(17,84,45),(18,85,46),(19,86,47),(20,87,48),(21,88,49),(22,89,50),(23,90,51),(24,91,52),(25,92,53),(26,93,54),(27,94,55),(28,95,56),(29,96,57),(30,97,58),(31,98,59),(32,99,60),(33,100,61),(34,101,62),(35,102,63),(36,103,64),(37,104,65),(38,105,66),(115,210,153),(116,211,154),(117,212,155),(118,213,156),(119,214,157),(120,215,158),(121,216,159),(122,217,160),(123,218,161),(124,219,162),(125,220,163),(126,221,164),(127,222,165),(128,223,166),(129,224,167),(130,225,168),(131,226,169),(132,227,170),(133,228,171),(134,191,172),(135,192,173),(136,193,174),(137,194,175),(138,195,176),(139,196,177),(140,197,178),(141,198,179),(142,199,180),(143,200,181),(144,201,182),(145,202,183),(146,203,184),(147,204,185),(148,205,186),(149,206,187),(150,207,188),(151,208,189),(152,209,190)], [(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)], [(1,115,20,134),(2,152,21,133),(3,151,22,132),(4,150,23,131),(5,149,24,130),(6,148,25,129),(7,147,26,128),(8,146,27,127),(9,145,28,126),(10,144,29,125),(11,143,30,124),(12,142,31,123),(13,141,32,122),(14,140,33,121),(15,139,34,120),(16,138,35,119),(17,137,36,118),(18,136,37,117),(19,135,38,116),(39,181,58,162),(40,180,59,161),(41,179,60,160),(42,178,61,159),(43,177,62,158),(44,176,63,157),(45,175,64,156),(46,174,65,155),(47,173,66,154),(48,172,67,153),(49,171,68,190),(50,170,69,189),(51,169,70,188),(52,168,71,187),(53,167,72,186),(54,166,73,185),(55,165,74,184),(56,164,75,183),(57,163,76,182),(77,201,96,220),(78,200,97,219),(79,199,98,218),(80,198,99,217),(81,197,100,216),(82,196,101,215),(83,195,102,214),(84,194,103,213),(85,193,104,212),(86,192,105,211),(87,191,106,210),(88,228,107,209),(89,227,108,208),(90,226,109,207),(91,225,110,206),(92,224,111,205),(93,223,112,204),(94,222,113,203),(95,221,114,202)]])
C3×Dic19 is a maximal subgroup of
D57⋊C4 C19⋊D12 C57⋊Q8 C12×D19
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 114A | ··· | 114R |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D19 | Dic19 | C3×D19 | C3×Dic19 |
| kernel | C3×Dic19 | C114 | Dic19 | C57 | C38 | C19 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 9 | 9 | 18 | 18 |
Matrix representation of C3×Dic19 ►in GL2(𝔽37) generated by
| 10 | 0 |
| 0 | 10 |
| 1 | 21 |
| 9 | 5 |
| 31 | 22 |
| 0 | 6 |
G:=sub<GL(2,GF(37))| [10,0,0,10],[1,9,21,5],[31,0,22,6] >;
C3×Dic19 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{19} % in TeX
G:=Group("C3xDic19"); // GroupNames label
G:=SmallGroup(228,4);
// by ID
G=gap.SmallGroup(228,4);
# by ID
G:=PCGroup([4,-2,-3,-2,-19,24,3459]);
// Polycyclic
G:=Group<a,b,c|a^3=b^38=1,c^2=b^19,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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