direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C20×D11, C220⋊4C2, C44⋊6C10, D22.2C10, C10.14D22, Dic11⋊5C10, C110.14C22, C55⋊8(C2×C4), C11⋊4(C2×C20), C2.1(C10×D11), C22.10(C2×C10), (C5×Dic11)⋊5C2, (C10×D11).2C2, SmallGroup(440,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C20×D11 |
Generators and relations for C20×D11
G = < a,b,c | a20=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C20×D11
►On 220 points
Generators in S220
(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)
(1 167 183 88 80 128 104 160 55 28 205)(2 168 184 89 61 129 105 141 56 29 206)(3 169 185 90 62 130 106 142 57 30 207)(4 170 186 91 63 131 107 143 58 31 208)(5 171 187 92 64 132 108 144 59 32 209)(6 172 188 93 65 133 109 145 60 33 210)(7 173 189 94 66 134 110 146 41 34 211)(8 174 190 95 67 135 111 147 42 35 212)(9 175 191 96 68 136 112 148 43 36 213)(10 176 192 97 69 137 113 149 44 37 214)(11 177 193 98 70 138 114 150 45 38 215)(12 178 194 99 71 139 115 151 46 39 216)(13 179 195 100 72 140 116 152 47 40 217)(14 180 196 81 73 121 117 153 48 21 218)(15 161 197 82 74 122 118 154 49 22 219)(16 162 198 83 75 123 119 155 50 23 220)(17 163 199 84 76 124 120 156 51 24 201)(18 164 200 85 77 125 101 157 52 25 202)(19 165 181 86 78 126 102 158 53 26 203)(20 166 182 87 79 127 103 159 54 27 204)
(1 205)(2 206)(3 207)(4 208)(5 209)(6 210)(7 211)(8 212)(9 213)(10 214)(11 215)(12 216)(13 217)(14 218)(15 219)(16 220)(17 201)(18 202)(19 203)(20 204)(21 180)(22 161)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 189)(42 190)(43 191)(44 192)(45 193)(46 194)(47 195)(48 196)(49 197)(50 198)(51 199)(52 200)(53 181)(54 182)(55 183)(56 184)(57 185)(58 186)(59 187)(60 188)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 101)(78 102)(79 103)(80 104)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)
G:=sub<Sym(220)| (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), (1,167,183,88,80,128,104,160,55,28,205)(2,168,184,89,61,129,105,141,56,29,206)(3,169,185,90,62,130,106,142,57,30,207)(4,170,186,91,63,131,107,143,58,31,208)(5,171,187,92,64,132,108,144,59,32,209)(6,172,188,93,65,133,109,145,60,33,210)(7,173,189,94,66,134,110,146,41,34,211)(8,174,190,95,67,135,111,147,42,35,212)(9,175,191,96,68,136,112,148,43,36,213)(10,176,192,97,69,137,113,149,44,37,214)(11,177,193,98,70,138,114,150,45,38,215)(12,178,194,99,71,139,115,151,46,39,216)(13,179,195,100,72,140,116,152,47,40,217)(14,180,196,81,73,121,117,153,48,21,218)(15,161,197,82,74,122,118,154,49,22,219)(16,162,198,83,75,123,119,155,50,23,220)(17,163,199,84,76,124,120,156,51,24,201)(18,164,200,85,77,125,101,157,52,25,202)(19,165,181,86,78,126,102,158,53,26,203)(20,166,182,87,79,127,103,159,54,27,204), (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,201)(18,202)(19,203)(20,204)(21,180)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,189)(42,190)(43,191)(44,192)(45,193)(46,194)(47,195)(48,196)(49,197)(50,198)(51,199)(52,200)(53,181)(54,182)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)>;
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), (1,167,183,88,80,128,104,160,55,28,205)(2,168,184,89,61,129,105,141,56,29,206)(3,169,185,90,62,130,106,142,57,30,207)(4,170,186,91,63,131,107,143,58,31,208)(5,171,187,92,64,132,108,144,59,32,209)(6,172,188,93,65,133,109,145,60,33,210)(7,173,189,94,66,134,110,146,41,34,211)(8,174,190,95,67,135,111,147,42,35,212)(9,175,191,96,68,136,112,148,43,36,213)(10,176,192,97,69,137,113,149,44,37,214)(11,177,193,98,70,138,114,150,45,38,215)(12,178,194,99,71,139,115,151,46,39,216)(13,179,195,100,72,140,116,152,47,40,217)(14,180,196,81,73,121,117,153,48,21,218)(15,161,197,82,74,122,118,154,49,22,219)(16,162,198,83,75,123,119,155,50,23,220)(17,163,199,84,76,124,120,156,51,24,201)(18,164,200,85,77,125,101,157,52,25,202)(19,165,181,86,78,126,102,158,53,26,203)(20,166,182,87,79,127,103,159,54,27,204), (1,205)(2,206)(3,207)(4,208)(5,209)(6,210)(7,211)(8,212)(9,213)(10,214)(11,215)(12,216)(13,217)(14,218)(15,219)(16,220)(17,201)(18,202)(19,203)(20,204)(21,180)(22,161)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,189)(42,190)(43,191)(44,192)(45,193)(46,194)(47,195)(48,196)(49,197)(50,198)(51,199)(52,200)(53,181)(54,182)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152) );
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)], [(1,167,183,88,80,128,104,160,55,28,205),(2,168,184,89,61,129,105,141,56,29,206),(3,169,185,90,62,130,106,142,57,30,207),(4,170,186,91,63,131,107,143,58,31,208),(5,171,187,92,64,132,108,144,59,32,209),(6,172,188,93,65,133,109,145,60,33,210),(7,173,189,94,66,134,110,146,41,34,211),(8,174,190,95,67,135,111,147,42,35,212),(9,175,191,96,68,136,112,148,43,36,213),(10,176,192,97,69,137,113,149,44,37,214),(11,177,193,98,70,138,114,150,45,38,215),(12,178,194,99,71,139,115,151,46,39,216),(13,179,195,100,72,140,116,152,47,40,217),(14,180,196,81,73,121,117,153,48,21,218),(15,161,197,82,74,122,118,154,49,22,219),(16,162,198,83,75,123,119,155,50,23,220),(17,163,199,84,76,124,120,156,51,24,201),(18,164,200,85,77,125,101,157,52,25,202),(19,165,181,86,78,126,102,158,53,26,203),(20,166,182,87,79,127,103,159,54,27,204)], [(1,205),(2,206),(3,207),(4,208),(5,209),(6,210),(7,211),(8,212),(9,213),(10,214),(11,215),(12,216),(13,217),(14,218),(15,219),(16,220),(17,201),(18,202),(19,203),(20,204),(21,180),(22,161),(23,162),(24,163),(25,164),(26,165),(27,166),(28,167),(29,168),(30,169),(31,170),(32,171),(33,172),(34,173),(35,174),(36,175),(37,176),(38,177),(39,178),(40,179),(41,189),(42,190),(43,191),(44,192),(45,193),(46,194),(47,195),(48,196),(49,197),(50,198),(51,199),(52,200),(53,181),(54,182),(55,183),(56,184),(57,185),(58,186),(59,187),(60,188),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,101),(78,102),(79,103),(80,104),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 11A | ··· | 11E | 20A | ··· | 20H | 20I | ··· | 20P | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 11 | ··· | 11 | 20 | ··· | 20 | 20 | ··· | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D11 | D22 | C4×D11 | C5×D11 | C10×D11 | C20×D11 |
| kernel | C20×D11 | C5×Dic11 | C220 | C10×D11 | C5×D11 | C4×D11 | Dic11 | C44 | D22 | D11 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 5 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C20×D11 ►in GL3(𝔽661) generated by
| 555 | 0 | 0 |
| 0 | 471 | 0 |
| 0 | 0 | 471 |
| 1 | 0 | 0 |
| 0 | 586 | 587 |
| 0 | 660 | 660 |
| 660 | 0 | 0 |
| 0 | 660 | 74 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(661))| [555,0,0,0,471,0,0,0,471],[1,0,0,0,586,660,0,587,660],[660,0,0,0,660,0,0,74,1] >;
C20×D11 in GAP, Magma, Sage, TeX
C_{20}\times D_{11} % in TeX
G:=Group("C20xD11"); // GroupNames label
G:=SmallGroup(440,25);
// by ID
G=gap.SmallGroup(440,25);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,106,10004]);
// Polycyclic
G:=Group<a,b,c|a^20=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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