direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C19×C3⋊D4, C57⋊9D4, D6⋊2C38, Dic3⋊C38, C38.17D6, C114.22C22, C3⋊2(D4×C19), (C2×C6)⋊2C38, (C2×C38)⋊3S3, (S3×C38)⋊5C2, (C2×C114)⋊6C2, C2.5(S3×C38), C6.5(C2×C38), C22⋊2(S3×C19), (Dic3×C19)⋊4C2, SmallGroup(456,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C19×C3⋊D4
G = < a,b,c,d | a19=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C19×C3⋊D4
►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)
(1 151 214)(2 152 215)(3 134 216)(4 135 217)(5 136 218)(6 137 219)(7 138 220)(8 139 221)(9 140 222)(10 141 223)(11 142 224)(12 143 225)(13 144 226)(14 145 227)(15 146 228)(16 147 210)(17 148 211)(18 149 212)(19 150 213)(20 89 73)(21 90 74)(22 91 75)(23 92 76)(24 93 58)(25 94 59)(26 95 60)(27 77 61)(28 78 62)(29 79 63)(30 80 64)(31 81 65)(32 82 66)(33 83 67)(34 84 68)(35 85 69)(36 86 70)(37 87 71)(38 88 72)(39 181 208)(40 182 209)(41 183 191)(42 184 192)(43 185 193)(44 186 194)(45 187 195)(46 188 196)(47 189 197)(48 190 198)(49 172 199)(50 173 200)(51 174 201)(52 175 202)(53 176 203)(54 177 204)(55 178 205)(56 179 206)(57 180 207)(96 161 115)(97 162 116)(98 163 117)(99 164 118)(100 165 119)(101 166 120)(102 167 121)(103 168 122)(104 169 123)(105 170 124)(106 171 125)(107 153 126)(108 154 127)(109 155 128)(110 156 129)(111 157 130)(112 158 131)(113 159 132)(114 160 133)
(1 109 199 24)(2 110 200 25)(3 111 201 26)(4 112 202 27)(5 113 203 28)(6 114 204 29)(7 96 205 30)(8 97 206 31)(9 98 207 32)(10 99 208 33)(11 100 209 34)(12 101 191 35)(13 102 192 36)(14 103 193 37)(15 104 194 38)(16 105 195 20)(17 106 196 21)(18 107 197 22)(19 108 198 23)(39 67 141 118)(40 68 142 119)(41 69 143 120)(42 70 144 121)(43 71 145 122)(44 72 146 123)(45 73 147 124)(46 74 148 125)(47 75 149 126)(48 76 150 127)(49 58 151 128)(50 59 152 129)(51 60 134 130)(52 61 135 131)(53 62 136 132)(54 63 137 133)(55 64 138 115)(56 65 139 116)(57 66 140 117)(77 217 158 175)(78 218 159 176)(79 219 160 177)(80 220 161 178)(81 221 162 179)(82 222 163 180)(83 223 164 181)(84 224 165 182)(85 225 166 183)(86 226 167 184)(87 227 168 185)(88 228 169 186)(89 210 170 187)(90 211 171 188)(91 212 153 189)(92 213 154 190)(93 214 155 172)(94 215 156 173)(95 216 157 174)
(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 153)(76 154)(77 131)(78 132)(79 133)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(134 216)(135 217)(136 218)(137 219)(138 220)(139 221)(140 222)(141 223)(142 224)(143 225)(144 226)(145 227)(146 228)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)
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), (1,151,214)(2,152,215)(3,134,216)(4,135,217)(5,136,218)(6,137,219)(7,138,220)(8,139,221)(9,140,222)(10,141,223)(11,142,224)(12,143,225)(13,144,226)(14,145,227)(15,146,228)(16,147,210)(17,148,211)(18,149,212)(19,150,213)(20,89,73)(21,90,74)(22,91,75)(23,92,76)(24,93,58)(25,94,59)(26,95,60)(27,77,61)(28,78,62)(29,79,63)(30,80,64)(31,81,65)(32,82,66)(33,83,67)(34,84,68)(35,85,69)(36,86,70)(37,87,71)(38,88,72)(39,181,208)(40,182,209)(41,183,191)(42,184,192)(43,185,193)(44,186,194)(45,187,195)(46,188,196)(47,189,197)(48,190,198)(49,172,199)(50,173,200)(51,174,201)(52,175,202)(53,176,203)(54,177,204)(55,178,205)(56,179,206)(57,180,207)(96,161,115)(97,162,116)(98,163,117)(99,164,118)(100,165,119)(101,166,120)(102,167,121)(103,168,122)(104,169,123)(105,170,124)(106,171,125)(107,153,126)(108,154,127)(109,155,128)(110,156,129)(111,157,130)(112,158,131)(113,159,132)(114,160,133), (1,109,199,24)(2,110,200,25)(3,111,201,26)(4,112,202,27)(5,113,203,28)(6,114,204,29)(7,96,205,30)(8,97,206,31)(9,98,207,32)(10,99,208,33)(11,100,209,34)(12,101,191,35)(13,102,192,36)(14,103,193,37)(15,104,194,38)(16,105,195,20)(17,106,196,21)(18,107,197,22)(19,108,198,23)(39,67,141,118)(40,68,142,119)(41,69,143,120)(42,70,144,121)(43,71,145,122)(44,72,146,123)(45,73,147,124)(46,74,148,125)(47,75,149,126)(48,76,150,127)(49,58,151,128)(50,59,152,129)(51,60,134,130)(52,61,135,131)(53,62,136,132)(54,63,137,133)(55,64,138,115)(56,65,139,116)(57,66,140,117)(77,217,158,175)(78,218,159,176)(79,219,160,177)(80,220,161,178)(81,221,162,179)(82,222,163,180)(83,223,164,181)(84,224,165,182)(85,225,166,183)(86,226,167,184)(87,227,168,185)(88,228,169,186)(89,210,170,187)(90,211,171,188)(91,212,153,189)(92,213,154,190)(93,214,155,172)(94,215,156,173)(95,216,157,174), (20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,153)(76,154)(77,131)(78,132)(79,133)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(134,216)(135,217)(136,218)(137,219)(138,220)(139,221)(140,222)(141,223)(142,224)(143,225)(144,226)(145,227)(146,228)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)>;
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), (1,151,214)(2,152,215)(3,134,216)(4,135,217)(5,136,218)(6,137,219)(7,138,220)(8,139,221)(9,140,222)(10,141,223)(11,142,224)(12,143,225)(13,144,226)(14,145,227)(15,146,228)(16,147,210)(17,148,211)(18,149,212)(19,150,213)(20,89,73)(21,90,74)(22,91,75)(23,92,76)(24,93,58)(25,94,59)(26,95,60)(27,77,61)(28,78,62)(29,79,63)(30,80,64)(31,81,65)(32,82,66)(33,83,67)(34,84,68)(35,85,69)(36,86,70)(37,87,71)(38,88,72)(39,181,208)(40,182,209)(41,183,191)(42,184,192)(43,185,193)(44,186,194)(45,187,195)(46,188,196)(47,189,197)(48,190,198)(49,172,199)(50,173,200)(51,174,201)(52,175,202)(53,176,203)(54,177,204)(55,178,205)(56,179,206)(57,180,207)(96,161,115)(97,162,116)(98,163,117)(99,164,118)(100,165,119)(101,166,120)(102,167,121)(103,168,122)(104,169,123)(105,170,124)(106,171,125)(107,153,126)(108,154,127)(109,155,128)(110,156,129)(111,157,130)(112,158,131)(113,159,132)(114,160,133), (1,109,199,24)(2,110,200,25)(3,111,201,26)(4,112,202,27)(5,113,203,28)(6,114,204,29)(7,96,205,30)(8,97,206,31)(9,98,207,32)(10,99,208,33)(11,100,209,34)(12,101,191,35)(13,102,192,36)(14,103,193,37)(15,104,194,38)(16,105,195,20)(17,106,196,21)(18,107,197,22)(19,108,198,23)(39,67,141,118)(40,68,142,119)(41,69,143,120)(42,70,144,121)(43,71,145,122)(44,72,146,123)(45,73,147,124)(46,74,148,125)(47,75,149,126)(48,76,150,127)(49,58,151,128)(50,59,152,129)(51,60,134,130)(52,61,135,131)(53,62,136,132)(54,63,137,133)(55,64,138,115)(56,65,139,116)(57,66,140,117)(77,217,158,175)(78,218,159,176)(79,219,160,177)(80,220,161,178)(81,221,162,179)(82,222,163,180)(83,223,164,181)(84,224,165,182)(85,225,166,183)(86,226,167,184)(87,227,168,185)(88,228,169,186)(89,210,170,187)(90,211,171,188)(91,212,153,189)(92,213,154,190)(93,214,155,172)(94,215,156,173)(95,216,157,174), (20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,153)(76,154)(77,131)(78,132)(79,133)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(134,216)(135,217)(136,218)(137,219)(138,220)(139,221)(140,222)(141,223)(142,224)(143,225)(144,226)(145,227)(146,228)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215) );
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)], [(1,151,214),(2,152,215),(3,134,216),(4,135,217),(5,136,218),(6,137,219),(7,138,220),(8,139,221),(9,140,222),(10,141,223),(11,142,224),(12,143,225),(13,144,226),(14,145,227),(15,146,228),(16,147,210),(17,148,211),(18,149,212),(19,150,213),(20,89,73),(21,90,74),(22,91,75),(23,92,76),(24,93,58),(25,94,59),(26,95,60),(27,77,61),(28,78,62),(29,79,63),(30,80,64),(31,81,65),(32,82,66),(33,83,67),(34,84,68),(35,85,69),(36,86,70),(37,87,71),(38,88,72),(39,181,208),(40,182,209),(41,183,191),(42,184,192),(43,185,193),(44,186,194),(45,187,195),(46,188,196),(47,189,197),(48,190,198),(49,172,199),(50,173,200),(51,174,201),(52,175,202),(53,176,203),(54,177,204),(55,178,205),(56,179,206),(57,180,207),(96,161,115),(97,162,116),(98,163,117),(99,164,118),(100,165,119),(101,166,120),(102,167,121),(103,168,122),(104,169,123),(105,170,124),(106,171,125),(107,153,126),(108,154,127),(109,155,128),(110,156,129),(111,157,130),(112,158,131),(113,159,132),(114,160,133)], [(1,109,199,24),(2,110,200,25),(3,111,201,26),(4,112,202,27),(5,113,203,28),(6,114,204,29),(7,96,205,30),(8,97,206,31),(9,98,207,32),(10,99,208,33),(11,100,209,34),(12,101,191,35),(13,102,192,36),(14,103,193,37),(15,104,194,38),(16,105,195,20),(17,106,196,21),(18,107,197,22),(19,108,198,23),(39,67,141,118),(40,68,142,119),(41,69,143,120),(42,70,144,121),(43,71,145,122),(44,72,146,123),(45,73,147,124),(46,74,148,125),(47,75,149,126),(48,76,150,127),(49,58,151,128),(50,59,152,129),(51,60,134,130),(52,61,135,131),(53,62,136,132),(54,63,137,133),(55,64,138,115),(56,65,139,116),(57,66,140,117),(77,217,158,175),(78,218,159,176),(79,219,160,177),(80,220,161,178),(81,221,162,179),(82,222,163,180),(83,223,164,181),(84,224,165,182),(85,225,166,183),(86,226,167,184),(87,227,168,185),(88,228,169,186),(89,210,170,187),(90,211,171,188),(91,212,153,189),(92,213,154,190),(93,214,155,172),(94,215,156,173),(95,216,157,174)], [(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,96),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,181),(40,182),(41,183),(42,184),(43,185),(44,186),(45,187),(46,188),(47,189),(48,190),(49,172),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,180),(58,155),(59,156),(60,157),(61,158),(62,159),(63,160),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,169),(73,170),(74,171),(75,153),(76,154),(77,131),(78,132),(79,133),(80,115),(81,116),(82,117),(83,118),(84,119),(85,120),(86,121),(87,122),(88,123),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(134,216),(135,217),(136,218),(137,219),(138,220),(139,221),(140,222),(141,223),(142,224),(143,225),(144,226),(145,227),(146,228),(147,210),(148,211),(149,212),(150,213),(151,214),(152,215)]])
171 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38AJ | 38AK | ··· | 38BB | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
171 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C19 | C38 | C38 | C38 | S3 | D4 | D6 | C3⋊D4 | S3×C19 | D4×C19 | S3×C38 | C19×C3⋊D4 |
| kernel | C19×C3⋊D4 | Dic3×C19 | S3×C38 | C2×C114 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C38 | C57 | C38 | C19 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 1 | 1 | 1 | 2 | 18 | 18 | 18 | 36 |
Matrix representation of C19×C3⋊D4 ►in GL2(𝔽229) generated by
| 27 | 0 |
| 0 | 27 |
| 228 | 228 |
| 1 | 0 |
| 166 | 103 |
| 166 | 63 |
| 1 | 0 |
| 228 | 228 |
G:=sub<GL(2,GF(229))| [27,0,0,27],[228,1,228,0],[166,166,103,63],[1,228,0,228] >;
C19×C3⋊D4 in GAP, Magma, Sage, TeX
C_{19}\times C_3\rtimes D_4 % in TeX
G:=Group("C19xC3:D4"); // GroupNames label
G:=SmallGroup(456,33);
// by ID
G=gap.SmallGroup(456,33);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-3,781,7604]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export