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G = C21×D11order 462 = 2·3·7·11

Direct product of C21 and D11

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

Aliases: C21×D11, C11⋊C42, C776C6, C332C14, C2315C2, SmallGroup(462,5)

Series: Derived Chief Lower central Upper central

C1C11 — C21×D11
C1C11C77C231 — C21×D11
C11 — C21×D11
C1C21

Generators and relations for C21×D11
 G = < a,b,c | a21=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C6
11C14
11C42

Smallest permutation representation of C21×D11
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 53 226 40 150 137 70 87 178 210 121)(2 54 227 41 151 138 71 88 179 190 122)(3 55 228 42 152 139 72 89 180 191 123)(4 56 229 22 153 140 73 90 181 192 124)(5 57 230 23 154 141 74 91 182 193 125)(6 58 231 24 155 142 75 92 183 194 126)(7 59 211 25 156 143 76 93 184 195 106)(8 60 212 26 157 144 77 94 185 196 107)(9 61 213 27 158 145 78 95 186 197 108)(10 62 214 28 159 146 79 96 187 198 109)(11 63 215 29 160 147 80 97 188 199 110)(12 43 216 30 161 127 81 98 189 200 111)(13 44 217 31 162 128 82 99 169 201 112)(14 45 218 32 163 129 83 100 170 202 113)(15 46 219 33 164 130 84 101 171 203 114)(16 47 220 34 165 131 64 102 172 204 115)(17 48 221 35 166 132 65 103 173 205 116)(18 49 222 36 167 133 66 104 174 206 117)(19 50 223 37 168 134 67 105 175 207 118)(20 51 224 38 148 135 68 85 176 208 119)(21 52 225 39 149 136 69 86 177 209 120)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 85)(39 86)(40 87)(41 88)(42 89)(43 200)(44 201)(45 202)(46 203)(47 204)(48 205)(49 206)(50 207)(51 208)(52 209)(53 210)(54 190)(55 191)(56 192)(57 193)(58 194)(59 195)(60 196)(61 197)(62 198)(63 199)(64 165)(65 166)(66 167)(67 168)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 161)(82 162)(83 163)(84 164)(169 217)(170 218)(171 219)(172 220)(173 221)(174 222)(175 223)(176 224)(177 225)(178 226)(179 227)(180 228)(181 229)(182 230)(183 231)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)

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

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

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

147 conjugacy classes

class 1  2 3A3B6A6B7A···7F11A···11E14A···14F21A···21L33A···33J42A···42L77A···77AD231A···231BH
order1233667···711···1114···1421···2133···3342···4277···77231···231
size1111111111···12···211···111···12···211···112···22···2

147 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C7C14C21C42D11C3×D11C7×D11C21×D11
kernelC21×D11C231C7×D11C77C3×D11C33D11C11C21C7C3C1
# reps11226612125103060

Matrix representation of C21×D11 in GL2(𝔽43) generated by

250
025
,
417
50
,
017
380
G:=sub<GL(2,GF(43))| [25,0,0,25],[4,5,17,0],[0,38,17,0] >;

C21×D11 in GAP, Magma, Sage, TeX

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

G:=Group("C21xD11");
// GroupNames label

G:=SmallGroup(462,5);
// by ID

G=gap.SmallGroup(462,5);
# by ID

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

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

Export

Subgroup lattice of C21×D11 in TeX

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