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G = C3×D73order 438 = 2·3·73

Direct product of C3 and D73

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

Aliases: C3×D73, C733C6, C2192C2, SmallGroup(438,4)

Series: Derived Chief Lower central Upper central

C1C73 — C3×D73
C1C73C219 — C3×D73
C73 — C3×D73
C1C3

Generators and relations for C3×D73
 G = < a,b,c | a3=b73=c2=1, ab=ba, ac=ca, cbc=b-1 >

73C2
73C6

Smallest permutation representation of C3×D73
On 219 points
Generators in S219
(1 184 75)(2 185 76)(3 186 77)(4 187 78)(5 188 79)(6 189 80)(7 190 81)(8 191 82)(9 192 83)(10 193 84)(11 194 85)(12 195 86)(13 196 87)(14 197 88)(15 198 89)(16 199 90)(17 200 91)(18 201 92)(19 202 93)(20 203 94)(21 204 95)(22 205 96)(23 206 97)(24 207 98)(25 208 99)(26 209 100)(27 210 101)(28 211 102)(29 212 103)(30 213 104)(31 214 105)(32 215 106)(33 216 107)(34 217 108)(35 218 109)(36 219 110)(37 147 111)(38 148 112)(39 149 113)(40 150 114)(41 151 115)(42 152 116)(43 153 117)(44 154 118)(45 155 119)(46 156 120)(47 157 121)(48 158 122)(49 159 123)(50 160 124)(51 161 125)(52 162 126)(53 163 127)(54 164 128)(55 165 129)(56 166 130)(57 167 131)(58 168 132)(59 169 133)(60 170 134)(61 171 135)(62 172 136)(63 173 137)(64 174 138)(65 175 139)(66 176 140)(67 177 141)(68 178 142)(69 179 143)(70 180 144)(71 181 145)(72 182 146)(73 183 74)
(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)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(74 75)(76 146)(77 145)(78 144)(79 143)(80 142)(81 141)(82 140)(83 139)(84 138)(85 137)(86 136)(87 135)(88 134)(89 133)(90 132)(91 131)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 122)(101 121)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(148 219)(149 218)(150 217)(151 216)(152 215)(153 214)(154 213)(155 212)(156 211)(157 210)(158 209)(159 208)(160 207)(161 206)(162 205)(163 204)(164 203)(165 202)(166 201)(167 200)(168 199)(169 198)(170 197)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)(183 184)

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

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

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

114 conjugacy classes

class 1  2 3A3B6A6B73A···73AJ219A···219BT
order12336673···73219···219
size1731173732···22···2

114 irreducible representations

dim111122
type+++
imageC1C2C3C6D73C3×D73
kernelC3×D73C219D73C73C3C1
# reps11223672

Matrix representation of C3×D73 in GL2(𝔽439) generated by

2670
0267
,
27428
43898
,
283316
180156
G:=sub<GL(2,GF(439))| [267,0,0,267],[27,438,428,98],[283,180,316,156] >;

C3×D73 in GAP, Magma, Sage, TeX

C_3\times D_{73}
% in TeX

G:=Group("C3xD73");
// GroupNames label

G:=SmallGroup(438,4);
// by ID

G=gap.SmallGroup(438,4);
# by ID

G:=PCGroup([3,-2,-3,-73,3890]);
// Polycyclic

G:=Group<a,b,c|a^3=b^73=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D73 in TeX

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