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G = D4×C53order 424 = 23·53

Direct product of C53 and D4

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

Aliases: D4×C53, C4⋊C106, C2123C2, C22⋊C106, C106.6C22, (C2×C106)⋊1C2, C2.1(C2×C106), SmallGroup(424,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C53
C1C2C106C2×C106 — D4×C53
C1C2 — D4×C53
C1C106 — D4×C53

Generators and relations for D4×C53
 G = < a,b,c | a53=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C106
2C106

Smallest permutation representation of D4×C53
On 212 points
Generators in S212
(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)
(1 57 177 124)(2 58 178 125)(3 59 179 126)(4 60 180 127)(5 61 181 128)(6 62 182 129)(7 63 183 130)(8 64 184 131)(9 65 185 132)(10 66 186 133)(11 67 187 134)(12 68 188 135)(13 69 189 136)(14 70 190 137)(15 71 191 138)(16 72 192 139)(17 73 193 140)(18 74 194 141)(19 75 195 142)(20 76 196 143)(21 77 197 144)(22 78 198 145)(23 79 199 146)(24 80 200 147)(25 81 201 148)(26 82 202 149)(27 83 203 150)(28 84 204 151)(29 85 205 152)(30 86 206 153)(31 87 207 154)(32 88 208 155)(33 89 209 156)(34 90 210 157)(35 91 211 158)(36 92 212 159)(37 93 160 107)(38 94 161 108)(39 95 162 109)(40 96 163 110)(41 97 164 111)(42 98 165 112)(43 99 166 113)(44 100 167 114)(45 101 168 115)(46 102 169 116)(47 103 170 117)(48 104 171 118)(49 105 172 119)(50 106 173 120)(51 54 174 121)(52 55 175 122)(53 56 176 123)
(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)

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

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

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

265 conjugacy classes

class 1 2A2B2C 4 53A···53AZ106A···106AZ106BA···106EZ212A···212AZ
order1222453···53106···106106···106212···212
size112221···11···12···22···2

265 irreducible representations

dim11111122
type++++
imageC1C2C2C53C106C106D4D4×C53
kernelD4×C53C212C2×C106D4C4C22C53C1
# reps1125252104152

Matrix representation of D4×C53 in GL2(𝔽1061) generated by

9780
0978
,
12
10601060
,
10
10601060
G:=sub<GL(2,GF(1061))| [978,0,0,978],[1,1060,2,1060],[1,1060,0,1060] >;

D4×C53 in GAP, Magma, Sage, TeX

D_4\times C_{53}
% in TeX

G:=Group("D4xC53");
// GroupNames label

G:=SmallGroup(424,10);
// by ID

G=gap.SmallGroup(424,10);
# by ID

G:=PCGroup([4,-2,-2,-53,-2,1713]);
// Polycyclic

G:=Group<a,b,c|a^53=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C53 in TeX

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