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## G = C22×D51order 408 = 23·3·17

### Direct product of C22 and D51

Aliases: C22×D51, C62D34, C342D6, C512C23, C1022C22, (C2×C34)⋊5S3, (C2×C6)⋊3D17, (C2×C102)⋊3C2, C172(C22×S3), C32(C22×D17), SmallGroup(408,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C51 — C22×D51
 Chief series C1 — C17 — C51 — D51 — D102 — C22×D51
 Lower central C51 — C22×D51
 Upper central C1 — C22

Generators and relations for C22×D51
G = < a,b,c,d | a2=b2=c51=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 812 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, D6, C2×C6, C17, C22×S3, D17, C34, C51, D34, C2×C34, D51, C102, C22×D17, D102, C2×C102, C22×D51
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, D17, D34, D51, C22×D17, D102, C22×D51

Smallest permutation representation of C22×D51
On 204 points
Generators in S204
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 189)(53 190)(54 191)(55 192)(56 193)(57 194)(58 195)(59 196)(60 197)(61 198)(62 199)(63 200)(64 201)(65 202)(66 203)(67 204)(68 154)(69 155)(70 156)(71 157)(72 158)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 169)(84 170)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 177)(92 178)(93 179)(94 180)(95 181)(96 182)(97 183)(98 184)(99 185)(100 186)(101 187)(102 188)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 169)(111 170)(112 171)(113 172)(114 173)(115 174)(116 175)(117 176)(118 177)(119 178)(120 179)(121 180)(122 181)(123 182)(124 183)(125 184)(126 185)(127 186)(128 187)(129 188)(130 189)(131 190)(132 191)(133 192)(134 193)(135 194)(136 195)(137 196)(138 197)(139 198)(140 199)(141 200)(142 201)(143 202)(144 203)(145 204)(146 154)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(153 161)
(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)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(100 102)(103 153)(104 152)(105 151)(106 150)(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)(154 169)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(170 204)(171 203)(172 202)(173 201)(174 200)(175 199)(176 198)(177 197)(178 196)(179 195)(180 194)(181 193)(182 192)(183 191)(184 190)(185 189)(186 188)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 17A ··· 17H 34A ··· 34X 51A ··· 51P 102A ··· 102AV order 1 2 2 2 2 2 2 2 3 6 6 6 17 ··· 17 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 1 1 51 51 51 51 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D17 D34 D51 D102 kernel C22×D51 D102 C2×C102 C2×C34 C34 C2×C6 C6 C22 C2 # reps 1 6 1 1 3 8 24 16 48

Matrix representation of C22×D51 in GL3(𝔽103) generated by

 1 0 0 0 102 0 0 0 102
,
 102 0 0 0 102 0 0 0 102
,
 1 0 0 0 37 97 0 6 77
,
 102 0 0 0 37 97 0 22 66
G:=sub<GL(3,GF(103))| [1,0,0,0,102,0,0,0,102],[102,0,0,0,102,0,0,0,102],[1,0,0,0,37,6,0,97,77],[102,0,0,0,37,22,0,97,66] >;

C22×D51 in GAP, Magma, Sage, TeX

C_2^2\times D_{51}
% in TeX

G:=Group("C2^2xD51");
// GroupNames label

G:=SmallGroup(408,45);
// by ID

G=gap.SmallGroup(408,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,323,9604]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^51=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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