direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×D19, C76⋊6C6, C228⋊4C2, D38.2C6, C6.14D38, Dic19⋊5C6, C114.14C22, C57⋊5(C2×C4), C19⋊4(C2×C12), C2.1(C6×D19), C38.10(C2×C6), (C6×D19).2C2, (C3×Dic19)⋊5C2, SmallGroup(456,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C12×D19 |
Generators and relations for C12×D19
G = < a,b,c | a12=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C12×D19
►On 228 points
Generators in S228
(1 210 63 130 94 172 36 203 41 143 107 155)(2 211 64 131 95 173 37 204 42 144 108 156)(3 212 65 132 77 174 38 205 43 145 109 157)(4 213 66 133 78 175 20 206 44 146 110 158)(5 214 67 115 79 176 21 207 45 147 111 159)(6 215 68 116 80 177 22 208 46 148 112 160)(7 216 69 117 81 178 23 209 47 149 113 161)(8 217 70 118 82 179 24 191 48 150 114 162)(9 218 71 119 83 180 25 192 49 151 96 163)(10 219 72 120 84 181 26 193 50 152 97 164)(11 220 73 121 85 182 27 194 51 134 98 165)(12 221 74 122 86 183 28 195 52 135 99 166)(13 222 75 123 87 184 29 196 53 136 100 167)(14 223 76 124 88 185 30 197 54 137 101 168)(15 224 58 125 89 186 31 198 55 138 102 169)(16 225 59 126 90 187 32 199 56 139 103 170)(17 226 60 127 91 188 33 200 57 140 104 171)(18 227 61 128 92 189 34 201 39 141 105 153)(19 228 62 129 93 190 35 202 40 142 106 154)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(33 38)(34 37)(35 36)(39 42)(40 41)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(58 67)(59 66)(60 65)(61 64)(62 63)(68 76)(69 75)(70 74)(71 73)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(92 95)(93 94)(96 98)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(115 125)(116 124)(117 123)(118 122)(119 121)(126 133)(127 132)(128 131)(129 130)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)(153 156)(154 155)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)(172 190)(173 189)(174 188)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)(191 195)(192 194)(196 209)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)(210 228)(211 227)(212 226)(213 225)(214 224)(215 223)(216 222)(217 221)(218 220)
G:=sub<Sym(228)| (1,210,63,130,94,172,36,203,41,143,107,155)(2,211,64,131,95,173,37,204,42,144,108,156)(3,212,65,132,77,174,38,205,43,145,109,157)(4,213,66,133,78,175,20,206,44,146,110,158)(5,214,67,115,79,176,21,207,45,147,111,159)(6,215,68,116,80,177,22,208,46,148,112,160)(7,216,69,117,81,178,23,209,47,149,113,161)(8,217,70,118,82,179,24,191,48,150,114,162)(9,218,71,119,83,180,25,192,49,151,96,163)(10,219,72,120,84,181,26,193,50,152,97,164)(11,220,73,121,85,182,27,194,51,134,98,165)(12,221,74,122,86,183,28,195,52,135,99,166)(13,222,75,123,87,184,29,196,53,136,100,167)(14,223,76,124,88,185,30,197,54,137,101,168)(15,224,58,125,89,186,31,198,55,138,102,169)(16,225,59,126,90,187,32,199,56,139,103,170)(17,226,60,127,91,188,33,200,57,140,104,171)(18,227,61,128,92,189,34,201,39,141,105,153)(19,228,62,129,93,190,35,202,40,142,106,154), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,38)(34,37)(35,36)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,67)(59,66)(60,65)(61,64)(62,63)(68,76)(69,75)(70,74)(71,73)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(92,95)(93,94)(96,98)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(115,125)(116,124)(117,123)(118,122)(119,121)(126,133)(127,132)(128,131)(129,130)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(153,156)(154,155)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,165)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(191,195)(192,194)(196,209)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,228)(211,227)(212,226)(213,225)(214,224)(215,223)(216,222)(217,221)(218,220)>;
G:=Group( (1,210,63,130,94,172,36,203,41,143,107,155)(2,211,64,131,95,173,37,204,42,144,108,156)(3,212,65,132,77,174,38,205,43,145,109,157)(4,213,66,133,78,175,20,206,44,146,110,158)(5,214,67,115,79,176,21,207,45,147,111,159)(6,215,68,116,80,177,22,208,46,148,112,160)(7,216,69,117,81,178,23,209,47,149,113,161)(8,217,70,118,82,179,24,191,48,150,114,162)(9,218,71,119,83,180,25,192,49,151,96,163)(10,219,72,120,84,181,26,193,50,152,97,164)(11,220,73,121,85,182,27,194,51,134,98,165)(12,221,74,122,86,183,28,195,52,135,99,166)(13,222,75,123,87,184,29,196,53,136,100,167)(14,223,76,124,88,185,30,197,54,137,101,168)(15,224,58,125,89,186,31,198,55,138,102,169)(16,225,59,126,90,187,32,199,56,139,103,170)(17,226,60,127,91,188,33,200,57,140,104,171)(18,227,61,128,92,189,34,201,39,141,105,153)(19,228,62,129,93,190,35,202,40,142,106,154), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(33,38)(34,37)(35,36)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,67)(59,66)(60,65)(61,64)(62,63)(68,76)(69,75)(70,74)(71,73)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(92,95)(93,94)(96,98)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(115,125)(116,124)(117,123)(118,122)(119,121)(126,133)(127,132)(128,131)(129,130)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(153,156)(154,155)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,165)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(191,195)(192,194)(196,209)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,228)(211,227)(212,226)(213,225)(214,224)(215,223)(216,222)(217,221)(218,220) );
G=PermutationGroup([[(1,210,63,130,94,172,36,203,41,143,107,155),(2,211,64,131,95,173,37,204,42,144,108,156),(3,212,65,132,77,174,38,205,43,145,109,157),(4,213,66,133,78,175,20,206,44,146,110,158),(5,214,67,115,79,176,21,207,45,147,111,159),(6,215,68,116,80,177,22,208,46,148,112,160),(7,216,69,117,81,178,23,209,47,149,113,161),(8,217,70,118,82,179,24,191,48,150,114,162),(9,218,71,119,83,180,25,192,49,151,96,163),(10,219,72,120,84,181,26,193,50,152,97,164),(11,220,73,121,85,182,27,194,51,134,98,165),(12,221,74,122,86,183,28,195,52,135,99,166),(13,222,75,123,87,184,29,196,53,136,100,167),(14,223,76,124,88,185,30,197,54,137,101,168),(15,224,58,125,89,186,31,198,55,138,102,169),(16,225,59,126,90,187,32,199,56,139,103,170),(17,226,60,127,91,188,33,200,57,140,104,171),(18,227,61,128,92,189,34,201,39,141,105,153),(19,228,62,129,93,190,35,202,40,142,106,154)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(33,38),(34,37),(35,36),(39,42),(40,41),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(58,67),(59,66),(60,65),(61,64),(62,63),(68,76),(69,75),(70,74),(71,73),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(92,95),(93,94),(96,98),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(115,125),(116,124),(117,123),(118,122),(119,121),(126,133),(127,132),(128,131),(129,130),(134,151),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,144),(142,143),(153,156),(154,155),(157,171),(158,170),(159,169),(160,168),(161,167),(162,166),(163,165),(172,190),(173,189),(174,188),(175,187),(176,186),(177,185),(178,184),(179,183),(180,182),(191,195),(192,194),(196,209),(197,208),(198,207),(199,206),(200,205),(201,204),(202,203),(210,228),(211,227),(212,226),(213,225),(214,224),(215,223),(216,222),(217,221),(218,220)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D19 | D38 | C3×D19 | C4×D19 | C6×D19 | C12×D19 |
| kernel | C12×D19 | C3×Dic19 | C228 | C6×D19 | C4×D19 | C3×D19 | Dic19 | C76 | D38 | D19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C12×D19 ►in GL2(𝔽37) generated by
| 23 | 0 |
| 0 | 23 |
| 36 | 33 |
| 7 | 27 |
| 27 | 30 |
| 30 | 10 |
G:=sub<GL(2,GF(37))| [23,0,0,23],[36,7,33,27],[27,30,30,10] >;
C12×D19 in GAP, Magma, Sage, TeX
C_{12}\times D_{19} % in TeX
G:=Group("C12xD19"); // GroupNames label
G:=SmallGroup(456,25);
// by ID
G=gap.SmallGroup(456,25);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,66,10804]);
// Polycyclic
G:=Group<a,b,c|a^12=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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