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## G = S3×C2×C38order 456 = 23·3·19

### Direct product of C2×C38 and S3

Aliases: S3×C2×C38, C574C23, C1144C22, C6⋊(C2×C38), C3⋊(C22×C38), (C2×C6)⋊3C38, (C2×C114)⋊7C2, SmallGroup(456,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C38
 Chief series C1 — C3 — C57 — S3×C19 — S3×C38 — S3×C2×C38
 Lower central C3 — S3×C2×C38
 Upper central C1 — C2×C38

Generators and relations for S3×C2×C38
G = < a,b,c,d | a2=b38=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, D6, C2×C6, C19, C22×S3, C38, C38, C57, C2×C38, C2×C38, S3×C19, C114, C22×C38, S3×C38, C2×C114, S3×C2×C38
Quotients: C1, C2, C22, S3, C23, D6, C19, C22×S3, C38, C2×C38, S3×C19, C22×C38, S3×C38, S3×C2×C38

Smallest permutation representation of S3×C2×C38
On 228 points
Generators in S228
(1 203)(2 204)(3 205)(4 206)(5 207)(6 208)(7 209)(8 210)(9 211)(10 212)(11 213)(12 214)(13 215)(14 216)(15 217)(16 218)(17 219)(18 220)(19 221)(20 222)(21 223)(22 224)(23 225)(24 226)(25 227)(26 228)(27 191)(28 192)(29 193)(30 194)(31 195)(32 196)(33 197)(34 198)(35 199)(36 200)(37 201)(38 202)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(115 165)(116 166)(117 167)(118 168)(119 169)(120 170)(121 171)(122 172)(123 173)(124 174)(125 175)(126 176)(127 177)(128 178)(129 179)(130 180)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 161)(150 162)(151 163)(152 164)
(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)
(1 109 187)(2 110 188)(3 111 189)(4 112 190)(5 113 153)(6 114 154)(7 77 155)(8 78 156)(9 79 157)(10 80 158)(11 81 159)(12 82 160)(13 83 161)(14 84 162)(15 85 163)(16 86 164)(17 87 165)(18 88 166)(19 89 167)(20 90 168)(21 91 169)(22 92 170)(23 93 171)(24 94 172)(25 95 173)(26 96 174)(27 97 175)(28 98 176)(29 99 177)(30 100 178)(31 101 179)(32 102 180)(33 103 181)(34 104 182)(35 105 183)(36 106 184)(37 107 185)(38 108 186)(39 124 228)(40 125 191)(41 126 192)(42 127 193)(43 128 194)(44 129 195)(45 130 196)(46 131 197)(47 132 198)(48 133 199)(49 134 200)(50 135 201)(51 136 202)(52 137 203)(53 138 204)(54 139 205)(55 140 206)(56 141 207)(57 142 208)(58 143 209)(59 144 210)(60 145 211)(61 146 212)(62 147 213)(63 148 214)(64 149 215)(65 150 216)(66 151 217)(67 152 218)(68 115 219)(69 116 220)(70 117 221)(71 118 222)(72 119 223)(73 120 224)(74 121 225)(75 122 226)(76 123 227)
(1 222)(2 223)(3 224)(4 225)(5 226)(6 227)(7 228)(8 191)(9 192)(10 193)(11 194)(12 195)(13 196)(14 197)(15 198)(16 199)(17 200)(18 201)(19 202)(20 203)(21 204)(22 205)(23 206)(24 207)(25 208)(26 209)(27 210)(28 211)(29 212)(30 213)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 221)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 167)(52 168)(53 169)(54 170)(55 171)(56 172)(57 173)(58 174)(59 175)(60 176)(61 177)(62 178)(63 179)(64 180)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 153)(76 154)(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)

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

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

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

228 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 19A ··· 19R 38A ··· 38BB 38BC ··· 38DV 57A ··· 57R 114A ··· 114BB order 1 2 2 2 2 2 2 2 3 6 6 6 19 ··· 19 38 ··· 38 38 ··· 38 57 ··· 57 114 ··· 114 size 1 1 1 1 3 3 3 3 2 2 2 2 1 ··· 1 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

228 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C19 C38 C38 S3 D6 S3×C19 S3×C38 kernel S3×C2×C38 S3×C38 C2×C114 C22×S3 D6 C2×C6 C2×C38 C38 C22 C2 # reps 1 6 1 18 108 18 1 3 18 54

Matrix representation of S3×C2×C38 in GL3(𝔽229) generated by

 228 0 0 0 1 0 0 0 1
,
 1 0 0 0 4 0 0 0 4
,
 1 0 0 0 0 228 0 1 228
,
 1 0 0 0 228 1 0 0 1
G:=sub<GL(3,GF(229))| [228,0,0,0,1,0,0,0,1],[1,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,228,228],[1,0,0,0,228,0,0,1,1] >;

S3×C2×C38 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{38}
% in TeX

G:=Group("S3xC2xC38");
// GroupNames label

G:=SmallGroup(456,52);
// by ID

G=gap.SmallGroup(456,52);
# by ID

G:=PCGroup([5,-2,-2,-2,-19,-3,7604]);
// Polycyclic

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

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𝔽