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## G = D4×C57order 456 = 23·3·19

### Direct product of C57 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C57, C4⋊C114, C767C6, C123C38, C2287C2, C222C114, C114.23C22, (C2×C38)⋊9C6, (C2×C6)⋊1C38, (C2×C114)⋊1C2, C6.6(C2×C38), C38.14(C2×C6), C2.1(C2×C114), SmallGroup(456,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C57
 Chief series C1 — C2 — C38 — C114 — C2×C114 — D4×C57
 Lower central C1 — C2 — D4×C57
 Upper central C1 — C114 — D4×C57

Generators and relations for D4×C57
G = < a,b,c | a57=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D4×C57
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 93 193 142)(2 94 194 143)(3 95 195 144)(4 96 196 145)(5 97 197 146)(6 98 198 147)(7 99 199 148)(8 100 200 149)(9 101 201 150)(10 102 202 151)(11 103 203 152)(12 104 204 153)(13 105 205 154)(14 106 206 155)(15 107 207 156)(16 108 208 157)(17 109 209 158)(18 110 210 159)(19 111 211 160)(20 112 212 161)(21 113 213 162)(22 114 214 163)(23 58 215 164)(24 59 216 165)(25 60 217 166)(26 61 218 167)(27 62 219 168)(28 63 220 169)(29 64 221 170)(30 65 222 171)(31 66 223 115)(32 67 224 116)(33 68 225 117)(34 69 226 118)(35 70 227 119)(36 71 228 120)(37 72 172 121)(38 73 173 122)(39 74 174 123)(40 75 175 124)(41 76 176 125)(42 77 177 126)(43 78 178 127)(44 79 179 128)(45 80 180 129)(46 81 181 130)(47 82 182 131)(48 83 183 132)(49 84 184 133)(50 85 185 134)(51 86 186 135)(52 87 187 136)(53 88 188 137)(54 89 189 138)(55 90 190 139)(56 91 191 140)(57 92 192 141)
(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 170)(65 171)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)

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

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

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

285 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 12A 12B 19A ··· 19R 38A ··· 38R 38S ··· 38BB 57A ··· 57AJ 76A ··· 76R 114A ··· 114AJ 114AK ··· 114DD 228A ··· 228AJ order 1 2 2 2 3 3 4 6 6 6 6 6 6 12 12 19 ··· 19 38 ··· 38 38 ··· 38 57 ··· 57 76 ··· 76 114 ··· 114 114 ··· 114 228 ··· 228 size 1 1 2 2 1 1 2 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

285 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C6 C6 C19 C38 C38 C57 C114 C114 D4 C3×D4 D4×C19 D4×C57 kernel D4×C57 C228 C2×C114 D4×C19 C76 C2×C38 C3×D4 C12 C2×C6 D4 C4 C22 C57 C19 C3 C1 # reps 1 1 2 2 2 4 18 18 36 36 36 72 1 2 18 36

Matrix representation of D4×C57 in GL3(𝔽229) generated by

 134 0 0 0 61 0 0 0 61
,
 228 0 0 0 101 63 0 227 128
,
 228 0 0 0 1 101 0 0 228
G:=sub<GL(3,GF(229))| [134,0,0,0,61,0,0,0,61],[228,0,0,0,101,227,0,63,128],[228,0,0,0,1,0,0,101,228] >;

D4×C57 in GAP, Magma, Sage, TeX

D_4\times C_{57}
% in TeX

G:=Group("D4xC57");
// GroupNames label

G:=SmallGroup(456,40);
// by ID

G=gap.SmallGroup(456,40);
# by ID

G:=PCGroup([5,-2,-2,-3,-19,-2,2301]);
// Polycyclic

G:=Group<a,b,c|a^57=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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