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## G = C11×D21order 462 = 2·3·7·11

### Direct product of C11 and D21

Aliases: C11×D21, C773S3, C333D7, C2314C2, C211C22, C7⋊(S3×C11), C3⋊(C11×D7), SmallGroup(462,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C11×D21
 Chief series C1 — C7 — C21 — C231 — C11×D21
 Lower central C21 — C11×D21
 Upper central C1 — C11

Generators and relations for C11×D21
G = < a,b,c | a11=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C11×D21
On 231 points
Generators in S231
(1 230 202 182 149 146 118 98 69 56 42)(2 231 203 183 150 147 119 99 70 57 22)(3 211 204 184 151 127 120 100 71 58 23)(4 212 205 185 152 128 121 101 72 59 24)(5 213 206 186 153 129 122 102 73 60 25)(6 214 207 187 154 130 123 103 74 61 26)(7 215 208 188 155 131 124 104 75 62 27)(8 216 209 189 156 132 125 105 76 63 28)(9 217 210 169 157 133 126 85 77 43 29)(10 218 190 170 158 134 106 86 78 44 30)(11 219 191 171 159 135 107 87 79 45 31)(12 220 192 172 160 136 108 88 80 46 32)(13 221 193 173 161 137 109 89 81 47 33)(14 222 194 174 162 138 110 90 82 48 34)(15 223 195 175 163 139 111 91 83 49 35)(16 224 196 176 164 140 112 92 84 50 36)(17 225 197 177 165 141 113 93 64 51 37)(18 226 198 178 166 142 114 94 65 52 38)(19 227 199 179 167 143 115 95 66 53 39)(20 228 200 180 168 144 116 96 67 54 40)(21 229 201 181 148 145 117 97 68 55 41)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 47)(44 46)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(64 73)(65 72)(66 71)(67 70)(68 69)(74 84)(75 83)(76 82)(77 81)(78 80)(85 89)(86 88)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(106 108)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(144 147)(145 146)(148 149)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(169 173)(170 172)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(190 192)(193 210)(194 209)(195 208)(196 207)(197 206)(198 205)(199 204)(200 203)(201 202)(211 227)(212 226)(213 225)(214 224)(215 223)(216 222)(217 221)(218 220)(228 231)(229 230)

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

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

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

132 conjugacy classes

 class 1 2 3 7A 7B 7C 11A ··· 11J 21A ··· 21F 22A ··· 22J 33A ··· 33J 77A ··· 77AD 231A ··· 231BH order 1 2 3 7 7 7 11 ··· 11 21 ··· 21 22 ··· 22 33 ··· 33 77 ··· 77 231 ··· 231 size 1 21 2 2 2 2 1 ··· 1 2 ··· 2 21 ··· 21 2 ··· 2 2 ··· 2 2 ··· 2

132 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C11 C22 S3 D7 D21 S3×C11 C11×D7 C11×D21 kernel C11×D21 C231 D21 C21 C77 C33 C11 C7 C3 C1 # reps 1 1 10 10 1 3 6 10 30 60

Matrix representation of C11×D21 in GL2(𝔽463) generated by

 362 0 0 362
,
 274 329 134 95
,
 95 171 329 368
G:=sub<GL(2,GF(463))| [362,0,0,362],[274,134,329,95],[95,329,171,368] >;

C11×D21 in GAP, Magma, Sage, TeX

C_{11}\times D_{21}
% in TeX

G:=Group("C11xD21");
// GroupNames label

G:=SmallGroup(462,10);
// by ID

G=gap.SmallGroup(462,10);
# by ID

G:=PCGroup([4,-2,-11,-3,-7,530,6339]);
// Polycyclic

G:=Group<a,b,c|a^11=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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