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G = C3×D79order 474 = 2·3·79

Direct product of C3 and D79

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

Aliases: C3×D79, C793C6, C2372C2, SmallGroup(474,4)

Series: Derived Chief Lower central Upper central

C1C79 — C3×D79
C1C79C237 — C3×D79
C79 — C3×D79
C1C3

Generators and relations for C3×D79
 G = < a,b,c | a3=b79=c2=1, ab=ba, ac=ca, cbc=b-1 >

79C2
79C6

Smallest permutation representation of C3×D79
On 237 points
Generators in S237
(1 165 119)(2 166 120)(3 167 121)(4 168 122)(5 169 123)(6 170 124)(7 171 125)(8 172 126)(9 173 127)(10 174 128)(11 175 129)(12 176 130)(13 177 131)(14 178 132)(15 179 133)(16 180 134)(17 181 135)(18 182 136)(19 183 137)(20 184 138)(21 185 139)(22 186 140)(23 187 141)(24 188 142)(25 189 143)(26 190 144)(27 191 145)(28 192 146)(29 193 147)(30 194 148)(31 195 149)(32 196 150)(33 197 151)(34 198 152)(35 199 153)(36 200 154)(37 201 155)(38 202 156)(39 203 157)(40 204 158)(41 205 80)(42 206 81)(43 207 82)(44 208 83)(45 209 84)(46 210 85)(47 211 86)(48 212 87)(49 213 88)(50 214 89)(51 215 90)(52 216 91)(53 217 92)(54 218 93)(55 219 94)(56 220 95)(57 221 96)(58 222 97)(59 223 98)(60 224 99)(61 225 100)(62 226 101)(63 227 102)(64 228 103)(65 229 104)(66 230 105)(67 231 106)(68 232 107)(69 233 108)(70 234 109)(71 235 110)(72 236 111)(73 237 112)(74 159 113)(75 160 114)(76 161 115)(77 162 116)(78 163 117)(79 164 118)
(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)(159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237)
(1 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(80 157)(81 156)(82 155)(83 154)(84 153)(85 152)(86 151)(87 150)(88 149)(89 148)(90 147)(91 146)(92 145)(93 144)(94 143)(95 142)(96 141)(97 140)(98 139)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(159 170)(160 169)(161 168)(162 167)(163 166)(164 165)(171 237)(172 236)(173 235)(174 234)(175 233)(176 232)(177 231)(178 230)(179 229)(180 228)(181 227)(182 226)(183 225)(184 224)(185 223)(186 222)(187 221)(188 220)(189 219)(190 218)(191 217)(192 216)(193 215)(194 214)(195 213)(196 212)(197 211)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)

G:=sub<Sym(237)| (1,165,119)(2,166,120)(3,167,121)(4,168,122)(5,169,123)(6,170,124)(7,171,125)(8,172,126)(9,173,127)(10,174,128)(11,175,129)(12,176,130)(13,177,131)(14,178,132)(15,179,133)(16,180,134)(17,181,135)(18,182,136)(19,183,137)(20,184,138)(21,185,139)(22,186,140)(23,187,141)(24,188,142)(25,189,143)(26,190,144)(27,191,145)(28,192,146)(29,193,147)(30,194,148)(31,195,149)(32,196,150)(33,197,151)(34,198,152)(35,199,153)(36,200,154)(37,201,155)(38,202,156)(39,203,157)(40,204,158)(41,205,80)(42,206,81)(43,207,82)(44,208,83)(45,209,84)(46,210,85)(47,211,86)(48,212,87)(49,213,88)(50,214,89)(51,215,90)(52,216,91)(53,217,92)(54,218,93)(55,219,94)(56,220,95)(57,221,96)(58,222,97)(59,223,98)(60,224,99)(61,225,100)(62,226,101)(63,227,102)(64,228,103)(65,229,104)(66,230,105)(67,231,106)(68,232,107)(69,233,108)(70,234,109)(71,235,110)(72,236,111)(73,237,112)(74,159,113)(75,160,114)(76,161,115)(77,162,116)(78,163,117)(79,164,118), (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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)(171,237)(172,236)(173,235)(174,234)(175,233)(176,232)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,225)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,218)(191,217)(192,216)(193,215)(194,214)(195,213)(196,212)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)>;

G:=Group( (1,165,119)(2,166,120)(3,167,121)(4,168,122)(5,169,123)(6,170,124)(7,171,125)(8,172,126)(9,173,127)(10,174,128)(11,175,129)(12,176,130)(13,177,131)(14,178,132)(15,179,133)(16,180,134)(17,181,135)(18,182,136)(19,183,137)(20,184,138)(21,185,139)(22,186,140)(23,187,141)(24,188,142)(25,189,143)(26,190,144)(27,191,145)(28,192,146)(29,193,147)(30,194,148)(31,195,149)(32,196,150)(33,197,151)(34,198,152)(35,199,153)(36,200,154)(37,201,155)(38,202,156)(39,203,157)(40,204,158)(41,205,80)(42,206,81)(43,207,82)(44,208,83)(45,209,84)(46,210,85)(47,211,86)(48,212,87)(49,213,88)(50,214,89)(51,215,90)(52,216,91)(53,217,92)(54,218,93)(55,219,94)(56,220,95)(57,221,96)(58,222,97)(59,223,98)(60,224,99)(61,225,100)(62,226,101)(63,227,102)(64,228,103)(65,229,104)(66,230,105)(67,231,106)(68,232,107)(69,233,108)(70,234,109)(71,235,110)(72,236,111)(73,237,112)(74,159,113)(75,160,114)(76,161,115)(77,162,116)(78,163,117)(79,164,118), (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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)(171,237)(172,236)(173,235)(174,234)(175,233)(176,232)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,225)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,218)(191,217)(192,216)(193,215)(194,214)(195,213)(196,212)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205) );

G=PermutationGroup([(1,165,119),(2,166,120),(3,167,121),(4,168,122),(5,169,123),(6,170,124),(7,171,125),(8,172,126),(9,173,127),(10,174,128),(11,175,129),(12,176,130),(13,177,131),(14,178,132),(15,179,133),(16,180,134),(17,181,135),(18,182,136),(19,183,137),(20,184,138),(21,185,139),(22,186,140),(23,187,141),(24,188,142),(25,189,143),(26,190,144),(27,191,145),(28,192,146),(29,193,147),(30,194,148),(31,195,149),(32,196,150),(33,197,151),(34,198,152),(35,199,153),(36,200,154),(37,201,155),(38,202,156),(39,203,157),(40,204,158),(41,205,80),(42,206,81),(43,207,82),(44,208,83),(45,209,84),(46,210,85),(47,211,86),(48,212,87),(49,213,88),(50,214,89),(51,215,90),(52,216,91),(53,217,92),(54,218,93),(55,219,94),(56,220,95),(57,221,96),(58,222,97),(59,223,98),(60,224,99),(61,225,100),(62,226,101),(63,227,102),(64,228,103),(65,229,104),(66,230,105),(67,231,106),(68,232,107),(69,233,108),(70,234,109),(71,235,110),(72,236,111),(73,237,112),(74,159,113),(75,160,114),(76,161,115),(77,162,116),(78,163,117),(79,164,118)], [(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),(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(80,157),(81,156),(82,155),(83,154),(84,153),(85,152),(86,151),(87,150),(88,149),(89,148),(90,147),(91,146),(92,145),(93,144),(94,143),(95,142),(96,141),(97,140),(98,139),(99,138),(100,137),(101,136),(102,135),(103,134),(104,133),(105,132),(106,131),(107,130),(108,129),(109,128),(110,127),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(159,170),(160,169),(161,168),(162,167),(163,166),(164,165),(171,237),(172,236),(173,235),(174,234),(175,233),(176,232),(177,231),(178,230),(179,229),(180,228),(181,227),(182,226),(183,225),(184,224),(185,223),(186,222),(187,221),(188,220),(189,219),(190,218),(191,217),(192,216),(193,215),(194,214),(195,213),(196,212),(197,211),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205)])

123 conjugacy classes

class 1  2 3A3B6A6B79A···79AM237A···237BZ
order12336679···79237···237
size1791179792···22···2

123 irreducible representations

dim111122
type+++
imageC1C2C3C6D79C3×D79
kernelC3×D79C237D79C79C3C1
# reps11223978

Matrix representation of C3×D79 in GL2(𝔽1423) generated by

7790
0779
,
9861
14220
,
01
10
G:=sub<GL(2,GF(1423))| [779,0,0,779],[986,1422,1,0],[0,1,1,0] >;

C3×D79 in GAP, Magma, Sage, TeX

C_3\times D_{79}
% in TeX

G:=Group("C3xD79");
// GroupNames label

G:=SmallGroup(474,4);
// by ID

G=gap.SmallGroup(474,4);
# by ID

G:=PCGroup([3,-2,-3,-79,4214]);
// Polycyclic

G:=Group<a,b,c|a^3=b^79=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D79 in TeX

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