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## G = S3×Dic19order 456 = 23·3·19

### Direct product of S3 and Dic19

Aliases: S3×Dic19, D6.D19, C6.2D38, C38.2D6, Dic573C2, C114.2C22, (S3×C19)⋊C4, C193(C4×S3), C572(C2×C4), (S3×C38).C2, C2.2(S3×D19), C31(C2×Dic19), (C3×Dic19)⋊1C2, SmallGroup(456,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C57 — S3×Dic19
 Chief series C1 — C19 — C57 — C114 — C3×Dic19 — S3×Dic19
 Lower central C57 — S3×Dic19
 Upper central C1 — C2

Generators and relations for S3×Dic19
G = < a,b,c,d | a3=b2=c38=1, d2=c19, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of S3×Dic19
On 228 points
Generators in S228
(1 95 123)(2 96 124)(3 97 125)(4 98 126)(5 99 127)(6 100 128)(7 101 129)(8 102 130)(9 103 131)(10 104 132)(11 105 133)(12 106 134)(13 107 135)(14 108 136)(15 109 137)(16 110 138)(17 111 139)(18 112 140)(19 113 141)(20 114 142)(21 77 143)(22 78 144)(23 79 145)(24 80 146)(25 81 147)(26 82 148)(27 83 149)(28 84 150)(29 85 151)(30 86 152)(31 87 115)(32 88 116)(33 89 117)(34 90 118)(35 91 119)(36 92 120)(37 93 121)(38 94 122)(39 173 208)(40 174 209)(41 175 210)(42 176 211)(43 177 212)(44 178 213)(45 179 214)(46 180 215)(47 181 216)(48 182 217)(49 183 218)(50 184 219)(51 185 220)(52 186 221)(53 187 222)(54 188 223)(55 189 224)(56 190 225)(57 153 226)(58 154 227)(59 155 228)(60 156 191)(61 157 192)(62 158 193)(63 159 194)(64 160 195)(65 161 196)(66 162 197)(67 163 198)(68 164 199)(69 165 200)(70 166 201)(71 167 202)(72 168 203)(73 169 204)(74 170 205)(75 171 206)(76 172 207)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 227)(40 228)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 201)(52 202)(53 203)(54 204)(55 205)(56 206)(57 207)(58 208)(59 209)(60 210)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 124)(78 125)(79 126)(80 127)(81 128)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 115)(107 116)(108 117)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(153 172)(154 173)(155 174)(156 175)(157 176)(158 177)(159 178)(160 179)(161 180)(162 181)(163 182)(164 183)(165 184)(166 185)(167 186)(168 187)(169 188)(170 189)(171 190)
(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)
(1 157 20 176)(2 156 21 175)(3 155 22 174)(4 154 23 173)(5 153 24 172)(6 190 25 171)(7 189 26 170)(8 188 27 169)(9 187 28 168)(10 186 29 167)(11 185 30 166)(12 184 31 165)(13 183 32 164)(14 182 33 163)(15 181 34 162)(16 180 35 161)(17 179 36 160)(18 178 37 159)(19 177 38 158)(39 126 58 145)(40 125 59 144)(41 124 60 143)(42 123 61 142)(43 122 62 141)(44 121 63 140)(45 120 64 139)(46 119 65 138)(47 118 66 137)(48 117 67 136)(49 116 68 135)(50 115 69 134)(51 152 70 133)(52 151 71 132)(53 150 72 131)(54 149 73 130)(55 148 74 129)(56 147 75 128)(57 146 76 127)(77 210 96 191)(78 209 97 228)(79 208 98 227)(80 207 99 226)(81 206 100 225)(82 205 101 224)(83 204 102 223)(84 203 103 222)(85 202 104 221)(86 201 105 220)(87 200 106 219)(88 199 107 218)(89 198 108 217)(90 197 109 216)(91 196 110 215)(92 195 111 214)(93 194 112 213)(94 193 113 212)(95 192 114 211)

G:=sub<Sym(228)| (1,95,123)(2,96,124)(3,97,125)(4,98,126)(5,99,127)(6,100,128)(7,101,129)(8,102,130)(9,103,131)(10,104,132)(11,105,133)(12,106,134)(13,107,135)(14,108,136)(15,109,137)(16,110,138)(17,111,139)(18,112,140)(19,113,141)(20,114,142)(21,77,143)(22,78,144)(23,79,145)(24,80,146)(25,81,147)(26,82,148)(27,83,149)(28,84,150)(29,85,151)(30,86,152)(31,87,115)(32,88,116)(33,89,117)(34,90,118)(35,91,119)(36,92,120)(37,93,121)(38,94,122)(39,173,208)(40,174,209)(41,175,210)(42,176,211)(43,177,212)(44,178,213)(45,179,214)(46,180,215)(47,181,216)(48,182,217)(49,183,218)(50,184,219)(51,185,220)(52,186,221)(53,187,222)(54,188,223)(55,189,224)(56,190,225)(57,153,226)(58,154,227)(59,155,228)(60,156,191)(61,157,192)(62,158,193)(63,159,194)(64,160,195)(65,161,196)(66,162,197)(67,163,198)(68,164,199)(69,165,200)(70,166,201)(71,167,202)(72,168,203)(73,169,204)(74,170,205)(75,171,206)(76,172,207), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,227)(40,228)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(153,172)(154,173)(155,174)(156,175)(157,176)(158,177)(159,178)(160,179)(161,180)(162,181)(163,182)(164,183)(165,184)(166,185)(167,186)(168,187)(169,188)(170,189)(171,190), (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), (1,157,20,176)(2,156,21,175)(3,155,22,174)(4,154,23,173)(5,153,24,172)(6,190,25,171)(7,189,26,170)(8,188,27,169)(9,187,28,168)(10,186,29,167)(11,185,30,166)(12,184,31,165)(13,183,32,164)(14,182,33,163)(15,181,34,162)(16,180,35,161)(17,179,36,160)(18,178,37,159)(19,177,38,158)(39,126,58,145)(40,125,59,144)(41,124,60,143)(42,123,61,142)(43,122,62,141)(44,121,63,140)(45,120,64,139)(46,119,65,138)(47,118,66,137)(48,117,67,136)(49,116,68,135)(50,115,69,134)(51,152,70,133)(52,151,71,132)(53,150,72,131)(54,149,73,130)(55,148,74,129)(56,147,75,128)(57,146,76,127)(77,210,96,191)(78,209,97,228)(79,208,98,227)(80,207,99,226)(81,206,100,225)(82,205,101,224)(83,204,102,223)(84,203,103,222)(85,202,104,221)(86,201,105,220)(87,200,106,219)(88,199,107,218)(89,198,108,217)(90,197,109,216)(91,196,110,215)(92,195,111,214)(93,194,112,213)(94,193,113,212)(95,192,114,211)>;

G:=Group( (1,95,123)(2,96,124)(3,97,125)(4,98,126)(5,99,127)(6,100,128)(7,101,129)(8,102,130)(9,103,131)(10,104,132)(11,105,133)(12,106,134)(13,107,135)(14,108,136)(15,109,137)(16,110,138)(17,111,139)(18,112,140)(19,113,141)(20,114,142)(21,77,143)(22,78,144)(23,79,145)(24,80,146)(25,81,147)(26,82,148)(27,83,149)(28,84,150)(29,85,151)(30,86,152)(31,87,115)(32,88,116)(33,89,117)(34,90,118)(35,91,119)(36,92,120)(37,93,121)(38,94,122)(39,173,208)(40,174,209)(41,175,210)(42,176,211)(43,177,212)(44,178,213)(45,179,214)(46,180,215)(47,181,216)(48,182,217)(49,183,218)(50,184,219)(51,185,220)(52,186,221)(53,187,222)(54,188,223)(55,189,224)(56,190,225)(57,153,226)(58,154,227)(59,155,228)(60,156,191)(61,157,192)(62,158,193)(63,159,194)(64,160,195)(65,161,196)(66,162,197)(67,163,198)(68,164,199)(69,165,200)(70,166,201)(71,167,202)(72,168,203)(73,169,204)(74,170,205)(75,171,206)(76,172,207), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,227)(40,228)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(153,172)(154,173)(155,174)(156,175)(157,176)(158,177)(159,178)(160,179)(161,180)(162,181)(163,182)(164,183)(165,184)(166,185)(167,186)(168,187)(169,188)(170,189)(171,190), (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), (1,157,20,176)(2,156,21,175)(3,155,22,174)(4,154,23,173)(5,153,24,172)(6,190,25,171)(7,189,26,170)(8,188,27,169)(9,187,28,168)(10,186,29,167)(11,185,30,166)(12,184,31,165)(13,183,32,164)(14,182,33,163)(15,181,34,162)(16,180,35,161)(17,179,36,160)(18,178,37,159)(19,177,38,158)(39,126,58,145)(40,125,59,144)(41,124,60,143)(42,123,61,142)(43,122,62,141)(44,121,63,140)(45,120,64,139)(46,119,65,138)(47,118,66,137)(48,117,67,136)(49,116,68,135)(50,115,69,134)(51,152,70,133)(52,151,71,132)(53,150,72,131)(54,149,73,130)(55,148,74,129)(56,147,75,128)(57,146,76,127)(77,210,96,191)(78,209,97,228)(79,208,98,227)(80,207,99,226)(81,206,100,225)(82,205,101,224)(83,204,102,223)(84,203,103,222)(85,202,104,221)(86,201,105,220)(87,200,106,219)(88,199,107,218)(89,198,108,217)(90,197,109,216)(91,196,110,215)(92,195,111,214)(93,194,112,213)(94,193,113,212)(95,192,114,211) );

G=PermutationGroup([(1,95,123),(2,96,124),(3,97,125),(4,98,126),(5,99,127),(6,100,128),(7,101,129),(8,102,130),(9,103,131),(10,104,132),(11,105,133),(12,106,134),(13,107,135),(14,108,136),(15,109,137),(16,110,138),(17,111,139),(18,112,140),(19,113,141),(20,114,142),(21,77,143),(22,78,144),(23,79,145),(24,80,146),(25,81,147),(26,82,148),(27,83,149),(28,84,150),(29,85,151),(30,86,152),(31,87,115),(32,88,116),(33,89,117),(34,90,118),(35,91,119),(36,92,120),(37,93,121),(38,94,122),(39,173,208),(40,174,209),(41,175,210),(42,176,211),(43,177,212),(44,178,213),(45,179,214),(46,180,215),(47,181,216),(48,182,217),(49,183,218),(50,184,219),(51,185,220),(52,186,221),(53,187,222),(54,188,223),(55,189,224),(56,190,225),(57,153,226),(58,154,227),(59,155,228),(60,156,191),(61,157,192),(62,158,193),(63,159,194),(64,160,195),(65,161,196),(66,162,197),(67,163,198),(68,164,199),(69,165,200),(70,166,201),(71,167,202),(72,168,203),(73,169,204),(74,170,205),(75,171,206),(76,172,207)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,227),(40,228),(41,191),(42,192),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,199),(50,200),(51,201),(52,202),(53,203),(54,204),(55,205),(56,206),(57,207),(58,208),(59,209),(60,210),(61,211),(62,212),(63,213),(64,214),(65,215),(66,216),(67,217),(68,218),(69,219),(70,220),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,124),(78,125),(79,126),(80,127),(81,128),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,115),(107,116),(108,117),(109,118),(110,119),(111,120),(112,121),(113,122),(114,123),(153,172),(154,173),(155,174),(156,175),(157,176),(158,177),(159,178),(160,179),(161,180),(162,181),(163,182),(164,183),(165,184),(166,185),(167,186),(168,187),(169,188),(170,189),(171,190)], [(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)], [(1,157,20,176),(2,156,21,175),(3,155,22,174),(4,154,23,173),(5,153,24,172),(6,190,25,171),(7,189,26,170),(8,188,27,169),(9,187,28,168),(10,186,29,167),(11,185,30,166),(12,184,31,165),(13,183,32,164),(14,182,33,163),(15,181,34,162),(16,180,35,161),(17,179,36,160),(18,178,37,159),(19,177,38,158),(39,126,58,145),(40,125,59,144),(41,124,60,143),(42,123,61,142),(43,122,62,141),(44,121,63,140),(45,120,64,139),(46,119,65,138),(47,118,66,137),(48,117,67,136),(49,116,68,135),(50,115,69,134),(51,152,70,133),(52,151,71,132),(53,150,72,131),(54,149,73,130),(55,148,74,129),(56,147,75,128),(57,146,76,127),(77,210,96,191),(78,209,97,228),(79,208,98,227),(80,207,99,226),(81,206,100,225),(82,205,101,224),(83,204,102,223),(84,203,103,222),(85,202,104,221),(86,201,105,220),(87,200,106,219),(88,199,107,218),(89,198,108,217),(90,197,109,216),(91,196,110,215),(92,195,111,214),(93,194,112,213),(94,193,113,212),(95,192,114,211)])

66 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 12A 12B 19A ··· 19I 38A ··· 38I 38J ··· 38AA 57A ··· 57I 114A ··· 114I order 1 2 2 2 3 4 4 4 4 6 12 12 19 ··· 19 38 ··· 38 38 ··· 38 57 ··· 57 114 ··· 114 size 1 1 3 3 2 19 19 57 57 2 38 38 2 ··· 2 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C4 S3 D6 C4×S3 D19 Dic19 D38 S3×D19 S3×Dic19 kernel S3×Dic19 C3×Dic19 Dic57 S3×C38 S3×C19 Dic19 C38 C19 D6 S3 C6 C2 C1 # reps 1 1 1 1 4 1 1 2 9 18 9 9 9

Matrix representation of S3×Dic19 in GL4(𝔽229) generated by

 1 0 0 0 0 1 0 0 0 0 227 70 0 0 121 1
,
 228 0 0 0 0 228 0 0 0 0 1 0 0 0 108 228
,
 0 228 0 0 1 127 0 0 0 0 1 0 0 0 0 1
,
 58 139 0 0 101 171 0 0 0 0 228 0 0 0 0 228
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,227,121,0,0,70,1],[228,0,0,0,0,228,0,0,0,0,1,108,0,0,0,228],[0,1,0,0,228,127,0,0,0,0,1,0,0,0,0,1],[58,101,0,0,139,171,0,0,0,0,228,0,0,0,0,228] >;

S3×Dic19 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{19}
% in TeX

G:=Group("S3xDic19");
// GroupNames label

G:=SmallGroup(456,13);
// by ID

G=gap.SmallGroup(456,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,20,168,10804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^38=1,d^2=c^19,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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