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G = C3×D77order 462 = 2·3·7·11

Direct product of C3 and D77

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

Aliases: C3×D77, C775C6, C332D7, C2312C2, C212D11, C11⋊(C3×D7), C73(C3×D11), SmallGroup(462,7)

Series: Derived Chief Lower central Upper central

C1C77 — C3×D77
C1C11C77C231 — C3×D77
C77 — C3×D77
C1C3

Generators and relations for C3×D77
 G = < a,b,c | a3=b77=c2=1, ab=ba, ac=ca, cbc=b-1 >

77C2
77C6
11D7
7D11
11C3×D7
7C3×D11

Smallest permutation representation of C3×D77
On 231 points
Generators in S231
(1 194 112)(2 195 113)(3 196 114)(4 197 115)(5 198 116)(6 199 117)(7 200 118)(8 201 119)(9 202 120)(10 203 121)(11 204 122)(12 205 123)(13 206 124)(14 207 125)(15 208 126)(16 209 127)(17 210 128)(18 211 129)(19 212 130)(20 213 131)(21 214 132)(22 215 133)(23 216 134)(24 217 135)(25 218 136)(26 219 137)(27 220 138)(28 221 139)(29 222 140)(30 223 141)(31 224 142)(32 225 143)(33 226 144)(34 227 145)(35 228 146)(36 229 147)(37 230 148)(38 231 149)(39 155 150)(40 156 151)(41 157 152)(42 158 153)(43 159 154)(44 160 78)(45 161 79)(46 162 80)(47 163 81)(48 164 82)(49 165 83)(50 166 84)(51 167 85)(52 168 86)(53 169 87)(54 170 88)(55 171 89)(56 172 90)(57 173 91)(58 174 92)(59 175 93)(60 176 94)(61 177 95)(62 178 96)(63 179 97)(64 180 98)(65 181 99)(66 182 100)(67 183 101)(68 184 102)(69 185 103)(70 186 104)(71 187 105)(72 188 106)(73 189 107)(74 190 108)(75 191 109)(76 192 110)(77 193 111)
(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)
(1 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(78 145)(79 144)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(146 154)(147 153)(148 152)(149 151)(156 231)(157 230)(158 229)(159 228)(160 227)(161 226)(162 225)(163 224)(164 223)(165 222)(166 221)(167 220)(168 219)(169 218)(170 217)(171 216)(172 215)(173 214)(174 213)(175 212)(176 211)(177 210)(178 209)(179 208)(180 207)(181 206)(182 205)(183 204)(184 203)(185 202)(186 201)(187 200)(188 199)(189 198)(190 197)(191 196)(192 195)(193 194)

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

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

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

120 conjugacy classes

class 1  2 3A3B6A6B7A7B7C11A···11E21A···21F33A···33J77A···77AD231A···231BH
order12336677711···1121···2133···3377···77231···231
size1771177772222···22···22···22···22···2

120 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6D7D11C3×D7C3×D11D77C3×D77
kernelC3×D77C231D77C77C33C21C11C7C3C1
# reps1122356103060

Matrix representation of C3×D77 in GL3(𝔽463) generated by

44100
010
001
,
100
015223
033144
,
46200
0202373
0191261
G:=sub<GL(3,GF(463))| [441,0,0,0,1,0,0,0,1],[1,0,0,0,15,331,0,223,44],[462,0,0,0,202,191,0,373,261] >;

C3×D77 in GAP, Magma, Sage, TeX

C_3\times D_{77}
% in TeX

G:=Group("C3xD77");
// GroupNames label

G:=SmallGroup(462,7);
// by ID

G=gap.SmallGroup(462,7);
# by ID

G:=PCGroup([4,-2,-3,-7,-11,434,6723]);
// Polycyclic

G:=Group<a,b,c|a^3=b^77=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D77 in TeX

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