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G = C4×D53order 424 = 23·53

Direct product of C4 and D53

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D53, C2122C2, C2.1D106, Dic532C2, D106.2C2, C106.2C22, C532(C2×C4), SmallGroup(424,5)

Series: Derived Chief Lower central Upper central

C1C53 — C4×D53
C1C53C106D106 — C4×D53
C53 — C4×D53
C1C4

Generators and relations for C4×D53
 G = < a,b,c | a4=b53=c2=1, ab=ba, ac=ca, cbc=b-1 >

53C2
53C2
53C22
53C4
53C2×C4

Smallest permutation representation of C4×D53
On 212 points
Generators in S212
(1 190 90 155)(2 191 91 156)(3 192 92 157)(4 193 93 158)(5 194 94 159)(6 195 95 107)(7 196 96 108)(8 197 97 109)(9 198 98 110)(10 199 99 111)(11 200 100 112)(12 201 101 113)(13 202 102 114)(14 203 103 115)(15 204 104 116)(16 205 105 117)(17 206 106 118)(18 207 54 119)(19 208 55 120)(20 209 56 121)(21 210 57 122)(22 211 58 123)(23 212 59 124)(24 160 60 125)(25 161 61 126)(26 162 62 127)(27 163 63 128)(28 164 64 129)(29 165 65 130)(30 166 66 131)(31 167 67 132)(32 168 68 133)(33 169 69 134)(34 170 70 135)(35 171 71 136)(36 172 72 137)(37 173 73 138)(38 174 74 139)(39 175 75 140)(40 176 76 141)(41 177 77 142)(42 178 78 143)(43 179 79 144)(44 180 80 145)(45 181 81 146)(46 182 82 147)(47 183 83 148)(48 184 84 149)(49 185 85 150)(50 186 86 151)(51 187 87 152)(52 188 88 153)(53 189 89 154)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(150 159)(151 158)(152 157)(153 156)(154 155)(160 166)(161 165)(162 164)(167 212)(168 211)(169 210)(170 209)(171 208)(172 207)(173 206)(174 205)(175 204)(176 203)(177 202)(178 201)(179 200)(180 199)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)

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

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

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

112 conjugacy classes

class 1 2A2B2C4A4B4C4D53A···53Z106A···106Z212A···212AZ
order1222444453···53106···106212···212
size1153531153532···22···22···2

112 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D53D106C4×D53
kernelC4×D53Dic53C212D106D53C4C2C1
# reps11114262652

Matrix representation of C4×D53 in GL3(𝔽1061) generated by

10300
010600
001060
,
100
0751
0295202
,
106000
0261968
0105800
G:=sub<GL(3,GF(1061))| [103,0,0,0,1060,0,0,0,1060],[1,0,0,0,75,295,0,1,202],[1060,0,0,0,261,105,0,968,800] >;

C4×D53 in GAP, Magma, Sage, TeX

C_4\times D_{53}
% in TeX

G:=Group("C4xD53");
// GroupNames label

G:=SmallGroup(424,5);
// by ID

G=gap.SmallGroup(424,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-53,21,6659]);
// Polycyclic

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

Export

Subgroup lattice of C4×D53 in TeX

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