direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C19, C3⋊C76, C57⋊3C4, C6.C38, C38.2S3, C114.3C2, C2.(S3×C19), SmallGroup(228,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C19 |
Generators and relations for Dic3×C19
G = < a,b,c | a19=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C19
►Regular action 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 66 207 210 160 39)(2 67 208 211 161 40)(3 68 209 212 162 41)(4 69 191 213 163 42)(5 70 192 214 164 43)(6 71 193 215 165 44)(7 72 194 216 166 45)(8 73 195 217 167 46)(9 74 196 218 168 47)(10 75 197 219 169 48)(11 76 198 220 170 49)(12 58 199 221 171 50)(13 59 200 222 153 51)(14 60 201 223 154 52)(15 61 202 224 155 53)(16 62 203 225 156 54)(17 63 204 226 157 55)(18 64 205 227 158 56)(19 65 206 228 159 57)(20 126 181 107 135 91)(21 127 182 108 136 92)(22 128 183 109 137 93)(23 129 184 110 138 94)(24 130 185 111 139 95)(25 131 186 112 140 77)(26 132 187 113 141 78)(27 133 188 114 142 79)(28 115 189 96 143 80)(29 116 190 97 144 81)(30 117 172 98 145 82)(31 118 173 99 146 83)(32 119 174 100 147 84)(33 120 175 101 148 85)(34 121 176 102 149 86)(35 122 177 103 150 87)(36 123 178 104 151 88)(37 124 179 105 152 89)(38 125 180 106 134 90)
(1 145 210 117)(2 146 211 118)(3 147 212 119)(4 148 213 120)(5 149 214 121)(6 150 215 122)(7 151 216 123)(8 152 217 124)(9 134 218 125)(10 135 219 126)(11 136 220 127)(12 137 221 128)(13 138 222 129)(14 139 223 130)(15 140 224 131)(16 141 225 132)(17 142 226 133)(18 143 227 115)(19 144 228 116)(20 75 107 169)(21 76 108 170)(22 58 109 171)(23 59 110 153)(24 60 111 154)(25 61 112 155)(26 62 113 156)(27 63 114 157)(28 64 96 158)(29 65 97 159)(30 66 98 160)(31 67 99 161)(32 68 100 162)(33 69 101 163)(34 70 102 164)(35 71 103 165)(36 72 104 166)(37 73 105 167)(38 74 106 168)(39 82 207 172)(40 83 208 173)(41 84 209 174)(42 85 191 175)(43 86 192 176)(44 87 193 177)(45 88 194 178)(46 89 195 179)(47 90 196 180)(48 91 197 181)(49 92 198 182)(50 93 199 183)(51 94 200 184)(52 95 201 185)(53 77 202 186)(54 78 203 187)(55 79 204 188)(56 80 205 189)(57 81 206 190)
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), (1,66,207,210,160,39)(2,67,208,211,161,40)(3,68,209,212,162,41)(4,69,191,213,163,42)(5,70,192,214,164,43)(6,71,193,215,165,44)(7,72,194,216,166,45)(8,73,195,217,167,46)(9,74,196,218,168,47)(10,75,197,219,169,48)(11,76,198,220,170,49)(12,58,199,221,171,50)(13,59,200,222,153,51)(14,60,201,223,154,52)(15,61,202,224,155,53)(16,62,203,225,156,54)(17,63,204,226,157,55)(18,64,205,227,158,56)(19,65,206,228,159,57)(20,126,181,107,135,91)(21,127,182,108,136,92)(22,128,183,109,137,93)(23,129,184,110,138,94)(24,130,185,111,139,95)(25,131,186,112,140,77)(26,132,187,113,141,78)(27,133,188,114,142,79)(28,115,189,96,143,80)(29,116,190,97,144,81)(30,117,172,98,145,82)(31,118,173,99,146,83)(32,119,174,100,147,84)(33,120,175,101,148,85)(34,121,176,102,149,86)(35,122,177,103,150,87)(36,123,178,104,151,88)(37,124,179,105,152,89)(38,125,180,106,134,90), (1,145,210,117)(2,146,211,118)(3,147,212,119)(4,148,213,120)(5,149,214,121)(6,150,215,122)(7,151,216,123)(8,152,217,124)(9,134,218,125)(10,135,219,126)(11,136,220,127)(12,137,221,128)(13,138,222,129)(14,139,223,130)(15,140,224,131)(16,141,225,132)(17,142,226,133)(18,143,227,115)(19,144,228,116)(20,75,107,169)(21,76,108,170)(22,58,109,171)(23,59,110,153)(24,60,111,154)(25,61,112,155)(26,62,113,156)(27,63,114,157)(28,64,96,158)(29,65,97,159)(30,66,98,160)(31,67,99,161)(32,68,100,162)(33,69,101,163)(34,70,102,164)(35,71,103,165)(36,72,104,166)(37,73,105,167)(38,74,106,168)(39,82,207,172)(40,83,208,173)(41,84,209,174)(42,85,191,175)(43,86,192,176)(44,87,193,177)(45,88,194,178)(46,89,195,179)(47,90,196,180)(48,91,197,181)(49,92,198,182)(50,93,199,183)(51,94,200,184)(52,95,201,185)(53,77,202,186)(54,78,203,187)(55,79,204,188)(56,80,205,189)(57,81,206,190)>;
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), (1,66,207,210,160,39)(2,67,208,211,161,40)(3,68,209,212,162,41)(4,69,191,213,163,42)(5,70,192,214,164,43)(6,71,193,215,165,44)(7,72,194,216,166,45)(8,73,195,217,167,46)(9,74,196,218,168,47)(10,75,197,219,169,48)(11,76,198,220,170,49)(12,58,199,221,171,50)(13,59,200,222,153,51)(14,60,201,223,154,52)(15,61,202,224,155,53)(16,62,203,225,156,54)(17,63,204,226,157,55)(18,64,205,227,158,56)(19,65,206,228,159,57)(20,126,181,107,135,91)(21,127,182,108,136,92)(22,128,183,109,137,93)(23,129,184,110,138,94)(24,130,185,111,139,95)(25,131,186,112,140,77)(26,132,187,113,141,78)(27,133,188,114,142,79)(28,115,189,96,143,80)(29,116,190,97,144,81)(30,117,172,98,145,82)(31,118,173,99,146,83)(32,119,174,100,147,84)(33,120,175,101,148,85)(34,121,176,102,149,86)(35,122,177,103,150,87)(36,123,178,104,151,88)(37,124,179,105,152,89)(38,125,180,106,134,90), (1,145,210,117)(2,146,211,118)(3,147,212,119)(4,148,213,120)(5,149,214,121)(6,150,215,122)(7,151,216,123)(8,152,217,124)(9,134,218,125)(10,135,219,126)(11,136,220,127)(12,137,221,128)(13,138,222,129)(14,139,223,130)(15,140,224,131)(16,141,225,132)(17,142,226,133)(18,143,227,115)(19,144,228,116)(20,75,107,169)(21,76,108,170)(22,58,109,171)(23,59,110,153)(24,60,111,154)(25,61,112,155)(26,62,113,156)(27,63,114,157)(28,64,96,158)(29,65,97,159)(30,66,98,160)(31,67,99,161)(32,68,100,162)(33,69,101,163)(34,70,102,164)(35,71,103,165)(36,72,104,166)(37,73,105,167)(38,74,106,168)(39,82,207,172)(40,83,208,173)(41,84,209,174)(42,85,191,175)(43,86,192,176)(44,87,193,177)(45,88,194,178)(46,89,195,179)(47,90,196,180)(48,91,197,181)(49,92,198,182)(50,93,199,183)(51,94,200,184)(52,95,201,185)(53,77,202,186)(54,78,203,187)(55,79,204,188)(56,80,205,189)(57,81,206,190) );
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)], [(1,66,207,210,160,39),(2,67,208,211,161,40),(3,68,209,212,162,41),(4,69,191,213,163,42),(5,70,192,214,164,43),(6,71,193,215,165,44),(7,72,194,216,166,45),(8,73,195,217,167,46),(9,74,196,218,168,47),(10,75,197,219,169,48),(11,76,198,220,170,49),(12,58,199,221,171,50),(13,59,200,222,153,51),(14,60,201,223,154,52),(15,61,202,224,155,53),(16,62,203,225,156,54),(17,63,204,226,157,55),(18,64,205,227,158,56),(19,65,206,228,159,57),(20,126,181,107,135,91),(21,127,182,108,136,92),(22,128,183,109,137,93),(23,129,184,110,138,94),(24,130,185,111,139,95),(25,131,186,112,140,77),(26,132,187,113,141,78),(27,133,188,114,142,79),(28,115,189,96,143,80),(29,116,190,97,144,81),(30,117,172,98,145,82),(31,118,173,99,146,83),(32,119,174,100,147,84),(33,120,175,101,148,85),(34,121,176,102,149,86),(35,122,177,103,150,87),(36,123,178,104,151,88),(37,124,179,105,152,89),(38,125,180,106,134,90)], [(1,145,210,117),(2,146,211,118),(3,147,212,119),(4,148,213,120),(5,149,214,121),(6,150,215,122),(7,151,216,123),(8,152,217,124),(9,134,218,125),(10,135,219,126),(11,136,220,127),(12,137,221,128),(13,138,222,129),(14,139,223,130),(15,140,224,131),(16,141,225,132),(17,142,226,133),(18,143,227,115),(19,144,228,116),(20,75,107,169),(21,76,108,170),(22,58,109,171),(23,59,110,153),(24,60,111,154),(25,61,112,155),(26,62,113,156),(27,63,114,157),(28,64,96,158),(29,65,97,159),(30,66,98,160),(31,67,99,161),(32,68,100,162),(33,69,101,163),(34,70,102,164),(35,71,103,165),(36,72,104,166),(37,73,105,167),(38,74,106,168),(39,82,207,172),(40,83,208,173),(41,84,209,174),(42,85,191,175),(43,86,192,176),(44,87,193,177),(45,88,194,178),(46,89,195,179),(47,90,196,180),(48,91,197,181),(49,92,198,182),(50,93,199,183),(51,94,200,184),(52,95,201,185),(53,77,202,186),(54,78,203,187),(55,79,204,188),(56,80,205,189),(57,81,206,190)]])
Dic3×C19 is a maximal subgroup of
D57⋊C4 C3⋊D76 C57⋊Q8 S3×C76
114 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57R | 76A | ··· | 76AJ | 114A | ··· | 114R |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C19 | C38 | C76 | S3 | Dic3 | S3×C19 | Dic3×C19 |
| kernel | Dic3×C19 | C114 | C57 | Dic3 | C6 | C3 | C38 | C19 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 1 | 1 | 18 | 18 |
Matrix representation of Dic3×C19 ►in GL2(𝔽229) generated by
| 27 | 0 |
| 0 | 27 |
| 0 | 228 |
| 1 | 1 |
| 189 | 13 |
| 53 | 40 |
G:=sub<GL(2,GF(229))| [27,0,0,27],[0,1,228,1],[189,53,13,40] >;
Dic3×C19 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{19} % in TeX
G:=Group("Dic3xC19"); // GroupNames label
G:=SmallGroup(228,3);
// by ID
G=gap.SmallGroup(228,3);
# by ID
G:=PCGroup([4,-2,-19,-2,-3,152,2435]);
// Polycyclic
G:=Group<a,b,c|a^19=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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