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G = S3×C77order 462 = 2·3·7·11

Direct product of C77 and S3

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

Aliases: S3×C77, C3⋊C154, C2317C2, C333C14, C213C22, SmallGroup(462,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C77
C1C3C33C231 — S3×C77
C3 — S3×C77
C1C77

Generators and relations for S3×C77
 G = < a,b,c | a77=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C14
3C22
3C154

Smallest permutation representation of S3×C77
On 231 points
Generators in S231
(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)
(1 225 133)(2 226 134)(3 227 135)(4 228 136)(5 229 137)(6 230 138)(7 231 139)(8 155 140)(9 156 141)(10 157 142)(11 158 143)(12 159 144)(13 160 145)(14 161 146)(15 162 147)(16 163 148)(17 164 149)(18 165 150)(19 166 151)(20 167 152)(21 168 153)(22 169 154)(23 170 78)(24 171 79)(25 172 80)(26 173 81)(27 174 82)(28 175 83)(29 176 84)(30 177 85)(31 178 86)(32 179 87)(33 180 88)(34 181 89)(35 182 90)(36 183 91)(37 184 92)(38 185 93)(39 186 94)(40 187 95)(41 188 96)(42 189 97)(43 190 98)(44 191 99)(45 192 100)(46 193 101)(47 194 102)(48 195 103)(49 196 104)(50 197 105)(51 198 106)(52 199 107)(53 200 108)(54 201 109)(55 202 110)(56 203 111)(57 204 112)(58 205 113)(59 206 114)(60 207 115)(61 208 116)(62 209 117)(63 210 118)(64 211 119)(65 212 120)(66 213 121)(67 214 122)(68 215 123)(69 216 124)(70 217 125)(71 218 126)(72 219 127)(73 220 128)(74 221 129)(75 222 130)(76 223 131)(77 224 132)
(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 189)(98 190)(99 191)(100 192)(101 193)(102 194)(103 195)(104 196)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)(113 205)(114 206)(115 207)(116 208)(117 209)(118 210)(119 211)(120 212)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 221)(130 222)(131 223)(132 224)(133 225)(134 226)(135 227)(136 228)(137 229)(138 230)(139 231)(140 155)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)(154 169)

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

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

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

231 conjugacy classes

class 1  2  3 7A···7F11A···11J14A···14F21A···21F22A···22J33A···33J77A···77BH154A···154BH231A···231BH
order1237···711···1114···1421···2122···2233···3377···77154···154231···231
size1321···11···13···32···23···32···21···13···32···2

231 irreducible representations

dim111111112222
type+++
imageC1C2C7C11C14C22C77C154S3S3×C7S3×C11S3×C77
kernelS3×C77C231S3×C11S3×C7C33C21S3C3C77C11C7C1
# reps116106106060161060

Matrix representation of S3×C77 in GL2(𝔽463) generated by

1590
0159
,
462462
10
,
10
462462
G:=sub<GL(2,GF(463))| [159,0,0,159],[462,1,462,0],[1,462,0,462] >;

S3×C77 in GAP, Magma, Sage, TeX

S_3\times C_{77}
% in TeX

G:=Group("S3xC77");
// GroupNames label

G:=SmallGroup(462,8);
// by ID

G=gap.SmallGroup(462,8);
# by ID

G:=PCGroup([4,-2,-7,-11,-3,4931]);
// Polycyclic

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

Export

Subgroup lattice of S3×C77 in TeX

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