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## G = D5×Dic11order 440 = 23·5·11

### Direct product of D5 and Dic11

Aliases: D5×Dic11, C10.2D22, C22.2D10, Dic553C2, D10.2D11, C110.2C22, C555(C2×C4), C113(C4×D5), (D5×C11)⋊1C4, C2.2(D5×D11), C52(C2×Dic11), (D5×C22).1C2, (C5×Dic11)⋊1C2, SmallGroup(440,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — D5×Dic11
 Chief series C1 — C11 — C55 — C110 — C5×Dic11 — D5×Dic11
 Lower central C55 — D5×Dic11
 Upper central C1 — C2

Generators and relations for D5×Dic11
G = < a,b,c,d | a5=b2=c22=1, d2=c11, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D5×Dic11
On 220 points
Generators in S220
(1 132 157 92 67)(2 111 158 93 68)(3 112 159 94 69)(4 113 160 95 70)(5 114 161 96 71)(6 115 162 97 72)(7 116 163 98 73)(8 117 164 99 74)(9 118 165 100 75)(10 119 166 101 76)(11 120 167 102 77)(12 121 168 103 78)(13 122 169 104 79)(14 123 170 105 80)(15 124 171 106 81)(16 125 172 107 82)(17 126 173 108 83)(18 127 174 109 84)(19 128 175 110 85)(20 129 176 89 86)(21 130 155 90 87)(22 131 156 91 88)(23 198 53 149 202)(24 177 54 150 203)(25 178 55 151 204)(26 179 56 152 205)(27 180 57 153 206)(28 181 58 154 207)(29 182 59 133 208)(30 183 60 134 209)(31 184 61 135 210)(32 185 62 136 211)(33 186 63 137 212)(34 187 64 138 213)(35 188 65 139 214)(36 189 66 140 215)(37 190 45 141 216)(38 191 46 142 217)(39 192 47 143 218)(40 193 48 144 219)(41 194 49 145 220)(42 195 50 146 199)(43 196 51 147 200)(44 197 52 148 201)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 211)(44 212)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 111)(105 112)(106 113)(107 114)(108 115)(109 116)(110 117)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 177)(140 178)(141 179)(142 180)(143 181)(144 182)(145 183)(146 184)(147 185)(148 186)(149 187)(150 188)(151 189)(152 190)(153 191)(154 192)(155 166)(156 167)(157 168)(158 169)(159 170)(160 171)(161 172)(162 173)(163 174)(164 175)(165 176)
(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 31 12 42)(2 30 13 41)(3 29 14 40)(4 28 15 39)(5 27 16 38)(6 26 17 37)(7 25 18 36)(8 24 19 35)(9 23 20 34)(10 44 21 33)(11 43 22 32)(45 162 56 173)(46 161 57 172)(47 160 58 171)(48 159 59 170)(49 158 60 169)(50 157 61 168)(51 156 62 167)(52 155 63 166)(53 176 64 165)(54 175 65 164)(55 174 66 163)(67 210 78 199)(68 209 79 220)(69 208 80 219)(70 207 81 218)(71 206 82 217)(72 205 83 216)(73 204 84 215)(74 203 85 214)(75 202 86 213)(76 201 87 212)(77 200 88 211)(89 138 100 149)(90 137 101 148)(91 136 102 147)(92 135 103 146)(93 134 104 145)(94 133 105 144)(95 154 106 143)(96 153 107 142)(97 152 108 141)(98 151 109 140)(99 150 110 139)(111 183 122 194)(112 182 123 193)(113 181 124 192)(114 180 125 191)(115 179 126 190)(116 178 127 189)(117 177 128 188)(118 198 129 187)(119 197 130 186)(120 196 131 185)(121 195 132 184)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 10A 10B 11A ··· 11E 20A 20B 20C 20D 22A ··· 22E 22F ··· 22O 55A ··· 55J 110A ··· 110J order 1 2 2 2 4 4 4 4 5 5 10 10 11 ··· 11 20 20 20 20 22 ··· 22 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 5 5 11 11 55 55 2 2 2 2 2 ··· 2 22 22 22 22 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 D10 D11 C4×D5 Dic11 D22 D5×D11 D5×Dic11 kernel D5×Dic11 C5×Dic11 Dic55 D5×C22 D5×C11 Dic11 C22 D10 C11 D5 C10 C2 C1 # reps 1 1 1 1 4 2 2 5 4 10 5 10 10

Matrix representation of D5×Dic11 in GL5(𝔽661)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 57 1 0 0 0 660 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 57 0 0 0 0 660
,
 660 0 0 0 0 0 643 1 0 0 0 172 174 0 0 0 0 0 1 0 0 0 0 0 1
,
 555 0 0 0 0 0 167 385 0 0 0 216 494 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(661))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,57,660,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,57,660],[660,0,0,0,0,0,643,172,0,0,0,1,174,0,0,0,0,0,1,0,0,0,0,0,1],[555,0,0,0,0,0,167,216,0,0,0,385,494,0,0,0,0,0,1,0,0,0,0,0,1] >;

D5×Dic11 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_{11}
% in TeX

G:=Group("D5xDic11");
// GroupNames label

G:=SmallGroup(440,18);
// by ID

G=gap.SmallGroup(440,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,20,328,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^22=1,d^2=c^11,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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