metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C55⋊3D4, C11⋊2D20, Dic11⋊D5, D110⋊4C2, D10⋊2D11, C22.6D10, C10.6D22, C110.6C22, (D5×C22)⋊2C2, C5⋊1(C11⋊D4), C2.6(D5×D11), (C5×Dic11)⋊3C2, SmallGroup(440,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11⋊D20
G = < a,b,c | a11=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C11⋊D20
►On 220 points
Generators in S220
(1 205 184 130 95 166 30 142 57 119 73)(2 74 120 58 143 31 167 96 131 185 206)(3 207 186 132 97 168 32 144 59 101 75)(4 76 102 60 145 33 169 98 133 187 208)(5 209 188 134 99 170 34 146 41 103 77)(6 78 104 42 147 35 171 100 135 189 210)(7 211 190 136 81 172 36 148 43 105 79)(8 80 106 44 149 37 173 82 137 191 212)(9 213 192 138 83 174 38 150 45 107 61)(10 62 108 46 151 39 175 84 139 193 214)(11 215 194 140 85 176 40 152 47 109 63)(12 64 110 48 153 21 177 86 121 195 216)(13 217 196 122 87 178 22 154 49 111 65)(14 66 112 50 155 23 179 88 123 197 218)(15 219 198 124 89 180 24 156 51 113 67)(16 68 114 52 157 25 161 90 125 199 220)(17 201 200 126 91 162 26 158 53 115 69)(18 70 116 54 159 27 163 92 127 181 202)(19 203 182 128 93 164 28 160 55 117 71)(20 72 118 56 141 29 165 94 129 183 204)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 179)(22 178)(23 177)(24 176)(25 175)(26 174)(27 173)(28 172)(29 171)(30 170)(31 169)(32 168)(33 167)(34 166)(35 165)(36 164)(37 163)(38 162)(39 161)(40 180)(41 130)(42 129)(43 128)(44 127)(45 126)(46 125)(47 124)(48 123)(49 122)(50 121)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 201)(62 220)(63 219)(64 218)(65 217)(66 216)(67 215)(68 214)(69 213)(70 212)(71 211)(72 210)(73 209)(74 208)(75 207)(76 206)(77 205)(78 204)(79 203)(80 202)(81 160)(82 159)(83 158)(84 157)(85 156)(86 155)(87 154)(88 153)(89 152)(90 151)(91 150)(92 149)(93 148)(94 147)(95 146)(96 145)(97 144)(98 143)(99 142)(100 141)(101 186)(102 185)(103 184)(104 183)(105 182)(106 181)(107 200)(108 199)(109 198)(110 197)(111 196)(112 195)(113 194)(114 193)(115 192)(116 191)(117 190)(118 189)(119 188)(120 187)
G:=sub<Sym(220)| (1,205,184,130,95,166,30,142,57,119,73)(2,74,120,58,143,31,167,96,131,185,206)(3,207,186,132,97,168,32,144,59,101,75)(4,76,102,60,145,33,169,98,133,187,208)(5,209,188,134,99,170,34,146,41,103,77)(6,78,104,42,147,35,171,100,135,189,210)(7,211,190,136,81,172,36,148,43,105,79)(8,80,106,44,149,37,173,82,137,191,212)(9,213,192,138,83,174,38,150,45,107,61)(10,62,108,46,151,39,175,84,139,193,214)(11,215,194,140,85,176,40,152,47,109,63)(12,64,110,48,153,21,177,86,121,195,216)(13,217,196,122,87,178,22,154,49,111,65)(14,66,112,50,155,23,179,88,123,197,218)(15,219,198,124,89,180,24,156,51,113,67)(16,68,114,52,157,25,161,90,125,199,220)(17,201,200,126,91,162,26,158,53,115,69)(18,70,116,54,159,27,163,92,127,181,202)(19,203,182,128,93,164,28,160,55,117,71)(20,72,118,56,141,29,165,94,129,183,204), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,180)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,201)(62,220)(63,219)(64,218)(65,217)(66,216)(67,215)(68,214)(69,213)(70,212)(71,211)(72,210)(73,209)(74,208)(75,207)(76,206)(77,205)(78,204)(79,203)(80,202)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,186)(102,185)(103,184)(104,183)(105,182)(106,181)(107,200)(108,199)(109,198)(110,197)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)>;
G:=Group( (1,205,184,130,95,166,30,142,57,119,73)(2,74,120,58,143,31,167,96,131,185,206)(3,207,186,132,97,168,32,144,59,101,75)(4,76,102,60,145,33,169,98,133,187,208)(5,209,188,134,99,170,34,146,41,103,77)(6,78,104,42,147,35,171,100,135,189,210)(7,211,190,136,81,172,36,148,43,105,79)(8,80,106,44,149,37,173,82,137,191,212)(9,213,192,138,83,174,38,150,45,107,61)(10,62,108,46,151,39,175,84,139,193,214)(11,215,194,140,85,176,40,152,47,109,63)(12,64,110,48,153,21,177,86,121,195,216)(13,217,196,122,87,178,22,154,49,111,65)(14,66,112,50,155,23,179,88,123,197,218)(15,219,198,124,89,180,24,156,51,113,67)(16,68,114,52,157,25,161,90,125,199,220)(17,201,200,126,91,162,26,158,53,115,69)(18,70,116,54,159,27,163,92,127,181,202)(19,203,182,128,93,164,28,160,55,117,71)(20,72,118,56,141,29,165,94,129,183,204), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,180)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,201)(62,220)(63,219)(64,218)(65,217)(66,216)(67,215)(68,214)(69,213)(70,212)(71,211)(72,210)(73,209)(74,208)(75,207)(76,206)(77,205)(78,204)(79,203)(80,202)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,186)(102,185)(103,184)(104,183)(105,182)(106,181)(107,200)(108,199)(109,198)(110,197)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187) );
G=PermutationGroup([[(1,205,184,130,95,166,30,142,57,119,73),(2,74,120,58,143,31,167,96,131,185,206),(3,207,186,132,97,168,32,144,59,101,75),(4,76,102,60,145,33,169,98,133,187,208),(5,209,188,134,99,170,34,146,41,103,77),(6,78,104,42,147,35,171,100,135,189,210),(7,211,190,136,81,172,36,148,43,105,79),(8,80,106,44,149,37,173,82,137,191,212),(9,213,192,138,83,174,38,150,45,107,61),(10,62,108,46,151,39,175,84,139,193,214),(11,215,194,140,85,176,40,152,47,109,63),(12,64,110,48,153,21,177,86,121,195,216),(13,217,196,122,87,178,22,154,49,111,65),(14,66,112,50,155,23,179,88,123,197,218),(15,219,198,124,89,180,24,156,51,113,67),(16,68,114,52,157,25,161,90,125,199,220),(17,201,200,126,91,162,26,158,53,115,69),(18,70,116,54,159,27,163,92,127,181,202),(19,203,182,128,93,164,28,160,55,117,71),(20,72,118,56,141,29,165,94,129,183,204)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,179),(22,178),(23,177),(24,176),(25,175),(26,174),(27,173),(28,172),(29,171),(30,170),(31,169),(32,168),(33,167),(34,166),(35,165),(36,164),(37,163),(38,162),(39,161),(40,180),(41,130),(42,129),(43,128),(44,127),(45,126),(46,125),(47,124),(48,123),(49,122),(50,121),(51,140),(52,139),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,201),(62,220),(63,219),(64,218),(65,217),(66,216),(67,215),(68,214),(69,213),(70,212),(71,211),(72,210),(73,209),(74,208),(75,207),(76,206),(77,205),(78,204),(79,203),(80,202),(81,160),(82,159),(83,158),(84,157),(85,156),(86,155),(87,154),(88,153),(89,152),(90,151),(91,150),(92,149),(93,148),(94,147),(95,146),(96,145),(97,144),(98,143),(99,142),(100,141),(101,186),(102,185),(103,184),(104,183),(105,182),(106,181),(107,200),(108,199),(109,198),(110,197),(111,196),(112,195),(113,194),(114,193),(115,192),(116,191),(117,190),(118,189),(119,188),(120,187)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 22F | ··· | 22O | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 10 | 110 | 22 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D11 | D20 | D22 | C11⋊D4 | D5×D11 | C11⋊D20 |
| kernel | C11⋊D20 | C5×Dic11 | D5×C22 | D110 | C55 | Dic11 | C22 | D10 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 10 | 10 |
Matrix representation of C11⋊D20 ►in GL4(𝔽661) generated by
| 0 | 1 | 0 | 0 |
| 660 | 538 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 660 | 0 | 0 | 0 |
| 123 | 1 | 0 | 0 |
| 0 | 0 | 268 | 226 |
| 0 | 0 | 302 | 341 |
| 1 | 0 | 0 | 0 |
| 538 | 660 | 0 | 0 |
| 0 | 0 | 1 | 125 |
| 0 | 0 | 0 | 660 |
G:=sub<GL(4,GF(661))| [0,660,0,0,1,538,0,0,0,0,1,0,0,0,0,1],[660,123,0,0,0,1,0,0,0,0,268,302,0,0,226,341],[1,538,0,0,0,660,0,0,0,0,1,0,0,0,125,660] >;
C11⋊D20 in GAP, Magma, Sage, TeX
C_{11}\rtimes D_{20} % in TeX
G:=Group("C11:D20"); // GroupNames label
G:=SmallGroup(440,22);
// by ID
G=gap.SmallGroup(440,22);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,20,61,328,10004]);
// Polycyclic
G:=Group<a,b,c|a^11=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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