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## G = C19×C3⋊D4order 456 = 23·3·19

### Direct product of C19 and C3⋊D4

Aliases: C19×C3⋊D4, C579D4, D62C38, Dic3⋊C38, C38.17D6, C114.22C22, C32(D4×C19), (C2×C6)⋊2C38, (C2×C38)⋊3S3, (S3×C38)⋊5C2, (C2×C114)⋊6C2, C2.5(S3×C38), C6.5(C2×C38), C222(S3×C19), (Dic3×C19)⋊4C2, SmallGroup(456,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C19×C3⋊D4
 Chief series C1 — C3 — C6 — C114 — S3×C38 — C19×C3⋊D4
 Lower central C3 — C6 — C19×C3⋊D4
 Upper central C1 — C38 — C2×C38

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

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

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

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

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

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