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## G = C2×C10×D11order 440 = 23·5·11

### Direct product of C2×C10 and D11

Aliases: C2×C10×D11, C553C23, C1103C22, (C2×C110)⋊5C2, C223(C2×C10), (C2×C22)⋊7C10, C113(C22×C10), SmallGroup(440,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×C10×D11
 Chief series C1 — C11 — C55 — C5×D11 — C10×D11 — C2×C10×D11
 Lower central C11 — C2×C10×D11
 Upper central C1 — C2×C10

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

Smallest permutation representation of 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

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