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## G = D5×C2×C22order 440 = 23·5·11

### Direct product of C2×C22 and D5

Aliases: D5×C2×C22, C554C23, C1104C22, C10⋊(C2×C22), C5⋊(C22×C22), (C2×C110)⋊7C2, (C2×C10)⋊3C22, SmallGroup(440,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C22
 Chief series C1 — C5 — C55 — D5×C11 — D5×C22 — D5×C2×C22
 Lower central C5 — D5×C2×C22
 Upper central C1 — C2×C22

Generators and relations for D5×C2×C22
G = < a,b,c,d | a2=b22=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 152 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C23, D5, C10, C11, D10, C2×C10, C22, C22, C22×D5, C2×C22, C2×C22, C55, C22×C22, D5×C11, C110, D5×C22, C2×C110, D5×C2×C22
Quotients: C1, C2, C22, C23, D5, C11, D10, C22, C22×D5, C2×C22, C22×C22, D5×C11, D5×C22, D5×C2×C22

Smallest permutation representation of D5×C2×C22
On 220 points
Generators in S220
(1 126)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 67)(44 68)(45 184)(46 185)(47 186)(48 187)(49 188)(50 189)(51 190)(52 191)(53 192)(54 193)(55 194)(56 195)(57 196)(58 197)(59 198)(60 177)(61 178)(62 179)(63 180)(64 181)(65 182)(66 183)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(155 210)(156 211)(157 212)(158 213)(159 214)(160 215)(161 216)(162 217)(163 218)(164 219)(165 220)(166 199)(167 200)(168 201)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)
(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 28 57 210 140)(2 29 58 211 141)(3 30 59 212 142)(4 31 60 213 143)(5 32 61 214 144)(6 33 62 215 145)(7 34 63 216 146)(8 35 64 217 147)(9 36 65 218 148)(10 37 66 219 149)(11 38 45 220 150)(12 39 46 199 151)(13 40 47 200 152)(14 41 48 201 153)(15 42 49 202 154)(16 43 50 203 133)(17 44 51 204 134)(18 23 52 205 135)(19 24 53 206 136)(20 25 54 207 137)(21 26 55 208 138)(22 27 56 209 139)(67 189 170 103 119)(68 190 171 104 120)(69 191 172 105 121)(70 192 173 106 122)(71 193 174 107 123)(72 194 175 108 124)(73 195 176 109 125)(74 196 155 110 126)(75 197 156 89 127)(76 198 157 90 128)(77 177 158 91 129)(78 178 159 92 130)(79 179 160 93 131)(80 180 161 94 132)(81 181 162 95 111)(82 182 163 96 112)(83 183 164 97 113)(84 184 165 98 114)(85 185 166 99 115)(86 186 167 100 116)(87 187 168 101 117)(88 188 169 102 118)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 167)(30 168)(31 169)(32 170)(33 171)(34 172)(35 173)(36 174)(37 175)(38 176)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 195)(46 196)(47 197)(48 198)(49 177)(50 178)(51 179)(52 180)(53 181)(54 182)(55 183)(56 184)(57 185)(58 186)(59 187)(60 188)(61 189)(62 190)(63 191)(64 192)(65 193)(66 194)(67 214)(68 215)(69 216)(70 217)(71 218)(72 219)(73 220)(74 199)(75 200)(76 201)(77 202)(78 203)(79 204)(80 205)(81 206)(82 207)(83 208)(84 209)(85 210)(86 211)(87 212)(88 213)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)(129 154)(130 133)(131 134)(132 135)

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

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

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

176 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 10A ··· 10F 11A ··· 11J 22A ··· 22AD 22AE ··· 22BR 55A ··· 55T 110A ··· 110BH order 1 2 2 2 2 2 2 2 5 5 10 ··· 10 11 ··· 11 22 ··· 22 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 1 1 5 5 5 5 2 2 2 ··· 2 1 ··· 1 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2

176 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C11 C22 C22 D5 D10 D5×C11 D5×C22 kernel D5×C2×C22 D5×C22 C2×C110 C22×D5 D10 C2×C10 C2×C22 C22 C22 C2 # reps 1 6 1 10 60 10 2 6 20 60

Matrix representation of D5×C2×C22 in GL3(𝔽331) generated by

 1 0 0 0 330 0 0 0 330
,
 330 0 0 0 80 0 0 0 80
,
 1 0 0 0 0 1 0 330 116
,
 1 0 0 0 0 330 0 330 0
G:=sub<GL(3,GF(331))| [1,0,0,0,330,0,0,0,330],[330,0,0,0,80,0,0,0,80],[1,0,0,0,0,330,0,1,116],[1,0,0,0,0,330,0,330,0] >;

D5×C2×C22 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{22}
% in TeX

G:=Group("D5xC2xC22");
// GroupNames label

G:=SmallGroup(440,49);
// by ID

G=gap.SmallGroup(440,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-11,-5,8804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^22=c^5=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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