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## G = Dic5×D11order 440 = 23·5·11

### Direct product of Dic5 and D11

Aliases: Dic5×D11, D22.D5, C10.1D22, C22.1D10, Dic552C2, C110.1C22, C554(C2×C4), C54(C4×D11), (C5×D11)⋊2C4, (C10×D11).C2, C2.1(D5×D11), C111(C2×Dic5), (C11×Dic5)⋊1C2, SmallGroup(440,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — Dic5×D11
 Chief series C1 — C11 — C55 — C110 — C10×D11 — Dic5×D11
 Lower central C55 — Dic5×D11
 Upper central C1 — C2

Generators and relations for Dic5×D11
G = < a,b,c,d | a10=c11=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of Dic5×D11
On 220 points
Generators in S220
(1 109 43 87 21 65 54 98 32 76)(2 110 44 88 22 66 55 99 33 77)(3 100 34 78 12 56 45 89 23 67)(4 101 35 79 13 57 46 90 24 68)(5 102 36 80 14 58 47 91 25 69)(6 103 37 81 15 59 48 92 26 70)(7 104 38 82 16 60 49 93 27 71)(8 105 39 83 17 61 50 94 28 72)(9 106 40 84 18 62 51 95 29 73)(10 107 41 85 19 63 52 96 30 74)(11 108 42 86 20 64 53 97 31 75)(111 177 133 199 155 166 122 188 144 210)(112 178 134 200 156 167 123 189 145 211)(113 179 135 201 157 168 124 190 146 212)(114 180 136 202 158 169 125 191 147 213)(115 181 137 203 159 170 126 192 148 214)(116 182 138 204 160 171 127 193 149 215)(117 183 139 205 161 172 128 194 150 216)(118 184 140 206 162 173 129 195 151 217)(119 185 141 207 163 174 130 196 152 218)(120 186 142 208 164 175 131 197 153 219)(121 187 143 209 165 176 132 198 154 220)
(1 175 65 120)(2 176 66 121)(3 166 56 111)(4 167 57 112)(5 168 58 113)(6 169 59 114)(7 170 60 115)(8 171 61 116)(9 172 62 117)(10 173 63 118)(11 174 64 119)(12 177 67 122)(13 178 68 123)(14 179 69 124)(15 180 70 125)(16 181 71 126)(17 182 72 127)(18 183 73 128)(19 184 74 129)(20 185 75 130)(21 186 76 131)(22 187 77 132)(23 188 78 133)(24 189 79 134)(25 190 80 135)(26 191 81 136)(27 192 82 137)(28 193 83 138)(29 194 84 139)(30 195 85 140)(31 196 86 141)(32 197 87 142)(33 198 88 143)(34 199 89 144)(35 200 90 145)(36 201 91 146)(37 202 92 147)(38 203 93 148)(39 204 94 149)(40 205 95 150)(41 206 96 151)(42 207 97 152)(43 208 98 153)(44 209 99 154)(45 210 100 155)(46 211 101 156)(47 212 102 157)(48 213 103 158)(49 214 104 159)(50 215 105 160)(51 216 106 161)(52 217 107 162)(53 218 108 163)(54 219 109 164)(55 220 110 165)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)(100 106)(101 105)(102 104)(107 110)(108 109)(111 117)(112 116)(113 115)(118 121)(119 120)(122 128)(123 127)(124 126)(129 132)(130 131)(133 139)(134 138)(135 137)(140 143)(141 142)(144 150)(145 149)(146 148)(151 154)(152 153)(155 161)(156 160)(157 159)(162 165)(163 164)(166 172)(167 171)(168 170)(173 176)(174 175)(177 183)(178 182)(179 181)(184 187)(185 186)(188 194)(189 193)(190 192)(195 198)(196 197)(199 205)(200 204)(201 203)(206 209)(207 208)(210 216)(211 215)(212 214)(217 220)(218 219)

G:=sub<Sym(220)| (1,109,43,87,21,65,54,98,32,76)(2,110,44,88,22,66,55,99,33,77)(3,100,34,78,12,56,45,89,23,67)(4,101,35,79,13,57,46,90,24,68)(5,102,36,80,14,58,47,91,25,69)(6,103,37,81,15,59,48,92,26,70)(7,104,38,82,16,60,49,93,27,71)(8,105,39,83,17,61,50,94,28,72)(9,106,40,84,18,62,51,95,29,73)(10,107,41,85,19,63,52,96,30,74)(11,108,42,86,20,64,53,97,31,75)(111,177,133,199,155,166,122,188,144,210)(112,178,134,200,156,167,123,189,145,211)(113,179,135,201,157,168,124,190,146,212)(114,180,136,202,158,169,125,191,147,213)(115,181,137,203,159,170,126,192,148,214)(116,182,138,204,160,171,127,193,149,215)(117,183,139,205,161,172,128,194,150,216)(118,184,140,206,162,173,129,195,151,217)(119,185,141,207,163,174,130,196,152,218)(120,186,142,208,164,175,131,197,153,219)(121,187,143,209,165,176,132,198,154,220), (1,175,65,120)(2,176,66,121)(3,166,56,111)(4,167,57,112)(5,168,58,113)(6,169,59,114)(7,170,60,115)(8,171,61,116)(9,172,62,117)(10,173,63,118)(11,174,64,119)(12,177,67,122)(13,178,68,123)(14,179,69,124)(15,180,70,125)(16,181,71,126)(17,182,72,127)(18,183,73,128)(19,184,74,129)(20,185,75,130)(21,186,76,131)(22,187,77,132)(23,188,78,133)(24,189,79,134)(25,190,80,135)(26,191,81,136)(27,192,82,137)(28,193,83,138)(29,194,84,139)(30,195,85,140)(31,196,86,141)(32,197,87,142)(33,198,88,143)(34,199,89,144)(35,200,90,145)(36,201,91,146)(37,202,92,147)(38,203,93,148)(39,204,94,149)(40,205,95,150)(41,206,96,151)(42,207,97,152)(43,208,98,153)(44,209,99,154)(45,210,100,155)(46,211,101,156)(47,212,102,157)(48,213,103,158)(49,214,104,159)(50,215,105,160)(51,216,106,161)(52,217,107,162)(53,218,108,163)(54,219,109,164)(55,220,110,165), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98)(100,106)(101,105)(102,104)(107,110)(108,109)(111,117)(112,116)(113,115)(118,121)(119,120)(122,128)(123,127)(124,126)(129,132)(130,131)(133,139)(134,138)(135,137)(140,143)(141,142)(144,150)(145,149)(146,148)(151,154)(152,153)(155,161)(156,160)(157,159)(162,165)(163,164)(166,172)(167,171)(168,170)(173,176)(174,175)(177,183)(178,182)(179,181)(184,187)(185,186)(188,194)(189,193)(190,192)(195,198)(196,197)(199,205)(200,204)(201,203)(206,209)(207,208)(210,216)(211,215)(212,214)(217,220)(218,219)>;

G:=Group( (1,109,43,87,21,65,54,98,32,76)(2,110,44,88,22,66,55,99,33,77)(3,100,34,78,12,56,45,89,23,67)(4,101,35,79,13,57,46,90,24,68)(5,102,36,80,14,58,47,91,25,69)(6,103,37,81,15,59,48,92,26,70)(7,104,38,82,16,60,49,93,27,71)(8,105,39,83,17,61,50,94,28,72)(9,106,40,84,18,62,51,95,29,73)(10,107,41,85,19,63,52,96,30,74)(11,108,42,86,20,64,53,97,31,75)(111,177,133,199,155,166,122,188,144,210)(112,178,134,200,156,167,123,189,145,211)(113,179,135,201,157,168,124,190,146,212)(114,180,136,202,158,169,125,191,147,213)(115,181,137,203,159,170,126,192,148,214)(116,182,138,204,160,171,127,193,149,215)(117,183,139,205,161,172,128,194,150,216)(118,184,140,206,162,173,129,195,151,217)(119,185,141,207,163,174,130,196,152,218)(120,186,142,208,164,175,131,197,153,219)(121,187,143,209,165,176,132,198,154,220), (1,175,65,120)(2,176,66,121)(3,166,56,111)(4,167,57,112)(5,168,58,113)(6,169,59,114)(7,170,60,115)(8,171,61,116)(9,172,62,117)(10,173,63,118)(11,174,64,119)(12,177,67,122)(13,178,68,123)(14,179,69,124)(15,180,70,125)(16,181,71,126)(17,182,72,127)(18,183,73,128)(19,184,74,129)(20,185,75,130)(21,186,76,131)(22,187,77,132)(23,188,78,133)(24,189,79,134)(25,190,80,135)(26,191,81,136)(27,192,82,137)(28,193,83,138)(29,194,84,139)(30,195,85,140)(31,196,86,141)(32,197,87,142)(33,198,88,143)(34,199,89,144)(35,200,90,145)(36,201,91,146)(37,202,92,147)(38,203,93,148)(39,204,94,149)(40,205,95,150)(41,206,96,151)(42,207,97,152)(43,208,98,153)(44,209,99,154)(45,210,100,155)(46,211,101,156)(47,212,102,157)(48,213,103,158)(49,214,104,159)(50,215,105,160)(51,216,106,161)(52,217,107,162)(53,218,108,163)(54,219,109,164)(55,220,110,165), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98)(100,106)(101,105)(102,104)(107,110)(108,109)(111,117)(112,116)(113,115)(118,121)(119,120)(122,128)(123,127)(124,126)(129,132)(130,131)(133,139)(134,138)(135,137)(140,143)(141,142)(144,150)(145,149)(146,148)(151,154)(152,153)(155,161)(156,160)(157,159)(162,165)(163,164)(166,172)(167,171)(168,170)(173,176)(174,175)(177,183)(178,182)(179,181)(184,187)(185,186)(188,194)(189,193)(190,192)(195,198)(196,197)(199,205)(200,204)(201,203)(206,209)(207,208)(210,216)(211,215)(212,214)(217,220)(218,219) );

G=PermutationGroup([[(1,109,43,87,21,65,54,98,32,76),(2,110,44,88,22,66,55,99,33,77),(3,100,34,78,12,56,45,89,23,67),(4,101,35,79,13,57,46,90,24,68),(5,102,36,80,14,58,47,91,25,69),(6,103,37,81,15,59,48,92,26,70),(7,104,38,82,16,60,49,93,27,71),(8,105,39,83,17,61,50,94,28,72),(9,106,40,84,18,62,51,95,29,73),(10,107,41,85,19,63,52,96,30,74),(11,108,42,86,20,64,53,97,31,75),(111,177,133,199,155,166,122,188,144,210),(112,178,134,200,156,167,123,189,145,211),(113,179,135,201,157,168,124,190,146,212),(114,180,136,202,158,169,125,191,147,213),(115,181,137,203,159,170,126,192,148,214),(116,182,138,204,160,171,127,193,149,215),(117,183,139,205,161,172,128,194,150,216),(118,184,140,206,162,173,129,195,151,217),(119,185,141,207,163,174,130,196,152,218),(120,186,142,208,164,175,131,197,153,219),(121,187,143,209,165,176,132,198,154,220)], [(1,175,65,120),(2,176,66,121),(3,166,56,111),(4,167,57,112),(5,168,58,113),(6,169,59,114),(7,170,60,115),(8,171,61,116),(9,172,62,117),(10,173,63,118),(11,174,64,119),(12,177,67,122),(13,178,68,123),(14,179,69,124),(15,180,70,125),(16,181,71,126),(17,182,72,127),(18,183,73,128),(19,184,74,129),(20,185,75,130),(21,186,76,131),(22,187,77,132),(23,188,78,133),(24,189,79,134),(25,190,80,135),(26,191,81,136),(27,192,82,137),(28,193,83,138),(29,194,84,139),(30,195,85,140),(31,196,86,141),(32,197,87,142),(33,198,88,143),(34,199,89,144),(35,200,90,145),(36,201,91,146),(37,202,92,147),(38,203,93,148),(39,204,94,149),(40,205,95,150),(41,206,96,151),(42,207,97,152),(43,208,98,153),(44,209,99,154),(45,210,100,155),(46,211,101,156),(47,212,102,157),(48,213,103,158),(49,214,104,159),(50,215,105,160),(51,216,106,161),(52,217,107,162),(53,218,108,163),(54,219,109,164),(55,220,110,165)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76),(78,84),(79,83),(80,82),(85,88),(86,87),(89,95),(90,94),(91,93),(96,99),(97,98),(100,106),(101,105),(102,104),(107,110),(108,109),(111,117),(112,116),(113,115),(118,121),(119,120),(122,128),(123,127),(124,126),(129,132),(130,131),(133,139),(134,138),(135,137),(140,143),(141,142),(144,150),(145,149),(146,148),(151,154),(152,153),(155,161),(156,160),(157,159),(162,165),(163,164),(166,172),(167,171),(168,170),(173,176),(174,175),(177,183),(178,182),(179,181),(184,187),(185,186),(188,194),(189,193),(190,192),(195,198),(196,197),(199,205),(200,204),(201,203),(206,209),(207,208),(210,216),(211,215),(212,214),(217,220),(218,219)]])

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 10A 10B 10C 10D 10E 10F 11A ··· 11E 22A ··· 22E 44A ··· 44J 55A ··· 55J 110A ··· 110J order 1 2 2 2 4 4 4 4 5 5 10 10 10 10 10 10 11 ··· 11 22 ··· 22 44 ··· 44 55 ··· 55 110 ··· 110 size 1 1 11 11 5 5 55 55 2 2 2 2 22 22 22 22 2 ··· 2 2 ··· 2 10 ··· 10 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + + + + - image C1 C2 C2 C2 C4 D5 Dic5 D10 D11 D22 C4×D11 D5×D11 Dic5×D11 kernel Dic5×D11 C11×Dic5 Dic55 C10×D11 C5×D11 D22 D11 C22 Dic5 C10 C5 C2 C1 # reps 1 1 1 1 4 2 4 2 5 5 10 10 10

Matrix representation of Dic5×D11 in GL4(𝔽661) generated by

 1 0 0 0 0 1 0 0 0 0 1 660 0 0 59 603
,
 1 0 0 0 0 1 0 0 0 0 411 327 0 0 577 250
,
 444 1 0 0 139 42 0 0 0 0 1 0 0 0 0 1
,
 42 660 0 0 441 619 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(661))| [1,0,0,0,0,1,0,0,0,0,1,59,0,0,660,603],[1,0,0,0,0,1,0,0,0,0,411,577,0,0,327,250],[444,139,0,0,1,42,0,0,0,0,1,0,0,0,0,1],[42,441,0,0,660,619,0,0,0,0,1,0,0,0,0,1] >;

Dic5×D11 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times D_{11}
% in TeX

G:=Group("Dic5xD11");
// GroupNames label

G:=SmallGroup(440,17);
// by ID

G=gap.SmallGroup(440,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,26,328,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^11=d^2=1,b^2=a^5,b*a*b^-1=a^-1,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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