direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C10×D11, C55⋊3C23, C110⋊3C22, (C2×C110)⋊5C2, C22⋊3(C2×C10), (C2×C22)⋊7C10, C11⋊3(C22×C10), SmallGroup(440,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C2×C10×D11 |
Generators and relations for C2×C10×D11
G = < a,b,c,d | a2=b10=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 284 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C23, C10, C10, C11, C2×C10, C2×C10, D11, C22, C22×C10, D22, C2×C22, C55, C22×D11, C5×D11, C110, C10×D11, C2×C110, C2×C10×D11
Quotients: C1, C2, C22, C5, C23, C10, C2×C10, D11, C22×C10, D22, C22×D11, C5×D11, C10×D11, C2×C10×D11
►On 220 points
Generators in S220
(1 175)(2 176)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 177)(13 178)(14 179)(15 180)(16 181)(17 182)(18 183)(19 184)(20 185)(21 186)(22 187)(23 188)(24 189)(25 190)(26 191)(27 192)(28 193)(29 194)(30 195)(31 196)(32 197)(33 198)(34 199)(35 200)(36 201)(37 202)(38 203)(39 204)(40 205)(41 206)(42 207)(43 208)(44 209)(45 210)(46 211)(47 212)(48 213)(49 214)(50 215)(51 216)(52 217)(53 218)(54 219)(55 220)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 164)(110 165)
(1 87 54 76 43 65 32 109 21 98)(2 88 55 77 44 66 33 110 22 99)(3 78 45 67 34 56 23 100 12 89)(4 79 46 68 35 57 24 101 13 90)(5 80 47 69 36 58 25 102 14 91)(6 81 48 70 37 59 26 103 15 92)(7 82 49 71 38 60 27 104 16 93)(8 83 50 72 39 61 28 105 17 94)(9 84 51 73 40 62 29 106 18 95)(10 85 52 74 41 63 30 107 19 96)(11 86 53 75 42 64 31 108 20 97)(111 188 155 177 144 166 133 210 122 199)(112 189 156 178 145 167 134 211 123 200)(113 190 157 179 146 168 135 212 124 201)(114 191 158 180 147 169 136 213 125 202)(115 192 159 181 148 170 137 214 126 203)(116 193 160 182 149 171 138 215 127 204)(117 194 161 183 150 172 139 216 128 205)(118 195 162 184 151 173 140 217 129 206)(119 196 163 185 152 174 141 218 130 207)(120 197 164 186 153 175 142 219 131 208)(121 198 165 187 154 176 143 220 132 209)
(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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 112)(9 111)(10 121)(11 120)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 132)(20 131)(21 130)(22 129)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 143)(31 142)(32 141)(33 140)(34 150)(35 149)(36 148)(37 147)(38 146)(39 145)(40 144)(41 154)(42 153)(43 152)(44 151)(45 161)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 165)(53 164)(54 163)(55 162)(56 172)(57 171)(58 170)(59 169)(60 168)(61 167)(62 166)(63 176)(64 175)(65 174)(66 173)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 187)(75 186)(76 185)(77 184)(78 194)(79 193)(80 192)(81 191)(82 190)(83 189)(84 188)(85 198)(86 197)(87 196)(88 195)(89 205)(90 204)(91 203)(92 202)(93 201)(94 200)(95 199)(96 209)(97 208)(98 207)(99 206)(100 216)(101 215)(102 214)(103 213)(104 212)(105 211)(106 210)(107 220)(108 219)(109 218)(110 217)
G:=sub<Sym(220)| (1,175)(2,176)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,177)(13,178)(14,179)(15,180)(16,181)(17,182)(18,183)(19,184)(20,185)(21,186)(22,187)(23,188)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165), (1,87,54,76,43,65,32,109,21,98)(2,88,55,77,44,66,33,110,22,99)(3,78,45,67,34,56,23,100,12,89)(4,79,46,68,35,57,24,101,13,90)(5,80,47,69,36,58,25,102,14,91)(6,81,48,70,37,59,26,103,15,92)(7,82,49,71,38,60,27,104,16,93)(8,83,50,72,39,61,28,105,17,94)(9,84,51,73,40,62,29,106,18,95)(10,85,52,74,41,63,30,107,19,96)(11,86,53,75,42,64,31,108,20,97)(111,188,155,177,144,166,133,210,122,199)(112,189,156,178,145,167,134,211,123,200)(113,190,157,179,146,168,135,212,124,201)(114,191,158,180,147,169,136,213,125,202)(115,192,159,181,148,170,137,214,126,203)(116,193,160,182,149,171,138,215,127,204)(117,194,161,183,150,172,139,216,128,205)(118,195,162,184,151,173,140,217,129,206)(119,196,163,185,152,174,141,218,130,207)(120,197,164,186,153,175,142,219,131,208)(121,198,165,187,154,176,143,220,132,209), (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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,121)(11,120)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,132)(20,131)(21,130)(22,129)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,143)(31,142)(32,141)(33,140)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,154)(42,153)(43,152)(44,151)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,165)(53,164)(54,163)(55,162)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,176)(64,175)(65,174)(66,173)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,187)(75,186)(76,185)(77,184)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,198)(86,197)(87,196)(88,195)(89,205)(90,204)(91,203)(92,202)(93,201)(94,200)(95,199)(96,209)(97,208)(98,207)(99,206)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,220)(108,219)(109,218)(110,217)>;
G:=Group( (1,175)(2,176)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,177)(13,178)(14,179)(15,180)(16,181)(17,182)(18,183)(19,184)(20,185)(21,186)(22,187)(23,188)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165), (1,87,54,76,43,65,32,109,21,98)(2,88,55,77,44,66,33,110,22,99)(3,78,45,67,34,56,23,100,12,89)(4,79,46,68,35,57,24,101,13,90)(5,80,47,69,36,58,25,102,14,91)(6,81,48,70,37,59,26,103,15,92)(7,82,49,71,38,60,27,104,16,93)(8,83,50,72,39,61,28,105,17,94)(9,84,51,73,40,62,29,106,18,95)(10,85,52,74,41,63,30,107,19,96)(11,86,53,75,42,64,31,108,20,97)(111,188,155,177,144,166,133,210,122,199)(112,189,156,178,145,167,134,211,123,200)(113,190,157,179,146,168,135,212,124,201)(114,191,158,180,147,169,136,213,125,202)(115,192,159,181,148,170,137,214,126,203)(116,193,160,182,149,171,138,215,127,204)(117,194,161,183,150,172,139,216,128,205)(118,195,162,184,151,173,140,217,129,206)(119,196,163,185,152,174,141,218,130,207)(120,197,164,186,153,175,142,219,131,208)(121,198,165,187,154,176,143,220,132,209), (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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,112)(9,111)(10,121)(11,120)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,132)(20,131)(21,130)(22,129)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,143)(31,142)(32,141)(33,140)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,154)(42,153)(43,152)(44,151)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,165)(53,164)(54,163)(55,162)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,176)(64,175)(65,174)(66,173)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,187)(75,186)(76,185)(77,184)(78,194)(79,193)(80,192)(81,191)(82,190)(83,189)(84,188)(85,198)(86,197)(87,196)(88,195)(89,205)(90,204)(91,203)(92,202)(93,201)(94,200)(95,199)(96,209)(97,208)(98,207)(99,206)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,220)(108,219)(109,218)(110,217) );
G=PermutationGroup([[(1,175),(2,176),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,177),(13,178),(14,179),(15,180),(16,181),(17,182),(18,183),(19,184),(20,185),(21,186),(22,187),(23,188),(24,189),(25,190),(26,191),(27,192),(28,193),(29,194),(30,195),(31,196),(32,197),(33,198),(34,199),(35,200),(36,201),(37,202),(38,203),(39,204),(40,205),(41,206),(42,207),(43,208),(44,209),(45,210),(46,211),(47,212),(48,213),(49,214),(50,215),(51,216),(52,217),(53,218),(54,219),(55,220),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155),(101,156),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(109,164),(110,165)], [(1,87,54,76,43,65,32,109,21,98),(2,88,55,77,44,66,33,110,22,99),(3,78,45,67,34,56,23,100,12,89),(4,79,46,68,35,57,24,101,13,90),(5,80,47,69,36,58,25,102,14,91),(6,81,48,70,37,59,26,103,15,92),(7,82,49,71,38,60,27,104,16,93),(8,83,50,72,39,61,28,105,17,94),(9,84,51,73,40,62,29,106,18,95),(10,85,52,74,41,63,30,107,19,96),(11,86,53,75,42,64,31,108,20,97),(111,188,155,177,144,166,133,210,122,199),(112,189,156,178,145,167,134,211,123,200),(113,190,157,179,146,168,135,212,124,201),(114,191,158,180,147,169,136,213,125,202),(115,192,159,181,148,170,137,214,126,203),(116,193,160,182,149,171,138,215,127,204),(117,194,161,183,150,172,139,216,128,205),(118,195,162,184,151,173,140,217,129,206),(119,196,163,185,152,174,141,218,130,207),(120,197,164,186,153,175,142,219,131,208),(121,198,165,187,154,176,143,220,132,209)], [(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)], [(1,119),(2,118),(3,117),(4,116),(5,115),(6,114),(7,113),(8,112),(9,111),(10,121),(11,120),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,132),(20,131),(21,130),(22,129),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,133),(30,143),(31,142),(32,141),(33,140),(34,150),(35,149),(36,148),(37,147),(38,146),(39,145),(40,144),(41,154),(42,153),(43,152),(44,151),(45,161),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,165),(53,164),(54,163),(55,162),(56,172),(57,171),(58,170),(59,169),(60,168),(61,167),(62,166),(63,176),(64,175),(65,174),(66,173),(67,183),(68,182),(69,181),(70,180),(71,179),(72,178),(73,177),(74,187),(75,186),(76,185),(77,184),(78,194),(79,193),(80,192),(81,191),(82,190),(83,189),(84,188),(85,198),(86,197),(87,196),(88,195),(89,205),(90,204),(91,203),(92,202),(93,201),(94,200),(95,199),(96,209),(97,208),(98,207),(99,206),(100,216),(101,215),(102,214),(103,213),(104,212),(105,211),(106,210),(107,220),(108,219),(109,218),(110,217)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 11A | ··· | 11E | 22A | ··· | 22O | 55A | ··· | 55T | 110A | ··· | 110BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D11 | D22 | C5×D11 | C10×D11 |
| kernel | C2×C10×D11 | C10×D11 | C2×C110 | C22×D11 | D22 | C2×C22 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 6 | 1 | 4 | 24 | 4 | 5 | 15 | 20 | 60 |
Matrix representation of C2×C10×D11 ►in GL3(𝔽331) generated by
| 330 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 330 | 0 | 0 |
| 0 | 181 | 0 |
| 0 | 0 | 181 |
| 1 | 0 | 0 |
| 0 | 123 | 1 |
| 0 | 330 | 0 |
| 330 | 0 | 0 |
| 0 | 0 | 330 |
| 0 | 330 | 0 |
G:=sub<GL(3,GF(331))| [330,0,0,0,1,0,0,0,1],[330,0,0,0,181,0,0,0,181],[1,0,0,0,123,330,0,1,0],[330,0,0,0,0,330,0,330,0] >;
C2×C10×D11 in GAP, Magma, Sage, TeX
C_2\times C_{10}\times D_{11} % in TeX
G:=Group("C2xC10xD11"); // GroupNames label
G:=SmallGroup(440,48);
// by ID
G=gap.SmallGroup(440,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,10004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations