direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C11×D20, C55⋊6D4, C44⋊3D5, C20⋊1C22, C220⋊5C2, D10⋊1C22, C22.15D10, C110.20C22, C4⋊(D5×C11), C5⋊1(D4×C11), (D5×C22)⋊4C2, C2.4(D5×C22), C10.3(C2×C22), SmallGroup(440,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×D20
G = < a,b,c | a11=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D20
►On 220 points
Generators in S220
(1 36 122 205 152 182 85 113 165 80 56)(2 37 123 206 153 183 86 114 166 61 57)(3 38 124 207 154 184 87 115 167 62 58)(4 39 125 208 155 185 88 116 168 63 59)(5 40 126 209 156 186 89 117 169 64 60)(6 21 127 210 157 187 90 118 170 65 41)(7 22 128 211 158 188 91 119 171 66 42)(8 23 129 212 159 189 92 120 172 67 43)(9 24 130 213 160 190 93 101 173 68 44)(10 25 131 214 141 191 94 102 174 69 45)(11 26 132 215 142 192 95 103 175 70 46)(12 27 133 216 143 193 96 104 176 71 47)(13 28 134 217 144 194 97 105 177 72 48)(14 29 135 218 145 195 98 106 178 73 49)(15 30 136 219 146 196 99 107 179 74 50)(16 31 137 220 147 197 100 108 180 75 51)(17 32 138 201 148 198 81 109 161 76 52)(18 33 139 202 149 199 82 110 162 77 53)(19 34 140 203 150 200 83 111 163 78 54)(20 35 121 204 151 181 84 112 164 79 55)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 140)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)(161 163)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)(181 197)(182 196)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)(198 200)(201 203)(204 220)(205 219)(206 218)(207 217)(208 216)(209 215)(210 214)(211 213)
G:=sub<Sym(220)| (1,36,122,205,152,182,85,113,165,80,56)(2,37,123,206,153,183,86,114,166,61,57)(3,38,124,207,154,184,87,115,167,62,58)(4,39,125,208,155,185,88,116,168,63,59)(5,40,126,209,156,186,89,117,169,64,60)(6,21,127,210,157,187,90,118,170,65,41)(7,22,128,211,158,188,91,119,171,66,42)(8,23,129,212,159,189,92,120,172,67,43)(9,24,130,213,160,190,93,101,173,68,44)(10,25,131,214,141,191,94,102,174,69,45)(11,26,132,215,142,192,95,103,175,70,46)(12,27,133,216,143,193,96,104,176,71,47)(13,28,134,217,144,194,97,105,177,72,48)(14,29,135,218,145,195,98,106,178,73,49)(15,30,136,219,146,196,99,107,179,74,50)(16,31,137,220,147,197,100,108,180,75,51)(17,32,138,201,148,198,81,109,161,76,52)(18,33,139,202,149,199,82,110,162,77,53)(19,34,140,203,150,200,83,111,163,78,54)(20,35,121,204,151,181,84,112,164,79,55), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,140)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,160)(161,163)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(181,197)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(198,200)(201,203)(204,220)(205,219)(206,218)(207,217)(208,216)(209,215)(210,214)(211,213)>;
G:=Group( (1,36,122,205,152,182,85,113,165,80,56)(2,37,123,206,153,183,86,114,166,61,57)(3,38,124,207,154,184,87,115,167,62,58)(4,39,125,208,155,185,88,116,168,63,59)(5,40,126,209,156,186,89,117,169,64,60)(6,21,127,210,157,187,90,118,170,65,41)(7,22,128,211,158,188,91,119,171,66,42)(8,23,129,212,159,189,92,120,172,67,43)(9,24,130,213,160,190,93,101,173,68,44)(10,25,131,214,141,191,94,102,174,69,45)(11,26,132,215,142,192,95,103,175,70,46)(12,27,133,216,143,193,96,104,176,71,47)(13,28,134,217,144,194,97,105,177,72,48)(14,29,135,218,145,195,98,106,178,73,49)(15,30,136,219,146,196,99,107,179,74,50)(16,31,137,220,147,197,100,108,180,75,51)(17,32,138,201,148,198,81,109,161,76,52)(18,33,139,202,149,199,82,110,162,77,53)(19,34,140,203,150,200,83,111,163,78,54)(20,35,121,204,151,181,84,112,164,79,55), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,140)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,160)(161,163)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(181,197)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(198,200)(201,203)(204,220)(205,219)(206,218)(207,217)(208,216)(209,215)(210,214)(211,213) );
G=PermutationGroup([[(1,36,122,205,152,182,85,113,165,80,56),(2,37,123,206,153,183,86,114,166,61,57),(3,38,124,207,154,184,87,115,167,62,58),(4,39,125,208,155,185,88,116,168,63,59),(5,40,126,209,156,186,89,117,169,64,60),(6,21,127,210,157,187,90,118,170,65,41),(7,22,128,211,158,188,91,119,171,66,42),(8,23,129,212,159,189,92,120,172,67,43),(9,24,130,213,160,190,93,101,173,68,44),(10,25,131,214,141,191,94,102,174,69,45),(11,26,132,215,142,192,95,103,175,70,46),(12,27,133,216,143,193,96,104,176,71,47),(13,28,134,217,144,194,97,105,177,72,48),(14,29,135,218,145,195,98,106,178,73,49),(15,30,136,219,146,196,99,107,179,74,50),(16,31,137,220,147,197,100,108,180,75,51),(17,32,138,201,148,198,81,109,161,76,52),(18,33,139,202,149,199,82,110,162,77,53),(19,34,140,203,150,200,83,111,163,78,54),(20,35,121,204,151,181,84,112,164,79,55)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(121,137),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(138,140),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150),(158,160),(161,163),(164,180),(165,179),(166,178),(167,177),(168,176),(169,175),(170,174),(171,173),(181,197),(182,196),(183,195),(184,194),(185,193),(186,192),(187,191),(188,190),(198,200),(201,203),(204,220),(205,219),(206,218),(207,217),(208,216),(209,215),(210,214),(211,213)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 11A | ··· | 11J | 20A | 20B | 20C | 20D | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | D5 | D10 | D20 | D4×C11 | D5×C11 | D5×C22 | C11×D20 |
| kernel | C11×D20 | C220 | D5×C22 | D20 | C20 | D10 | C55 | C44 | C22 | C11 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 2 | 2 | 4 | 10 | 20 | 20 | 40 |
Matrix representation of C11×D20 ►in GL2(𝔽661) generated by
| 68 | 0 |
| 0 | 68 |
| 572 | 624 |
| 37 | 409 |
| 0 | 660 |
| 660 | 0 |
G:=sub<GL(2,GF(661))| [68,0,0,68],[572,37,624,409],[0,660,660,0] >;
C11×D20 in GAP, Magma, Sage, TeX
C_{11}\times D_{20} % in TeX
G:=Group("C11xD20"); // GroupNames label
G:=SmallGroup(440,31);
// by ID
G=gap.SmallGroup(440,31);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-5,461,226,8804]);
// Polycyclic
G:=Group<a,b,c|a^11=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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