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G = C2×D83D5order 320 = 26·5

Direct product of C2 and D83D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D83D5, D812D10, C20.3C24, C40.32C23, Dic2013C22, Dic10.1C23, (C10×D8)⋊8C2, (C2×D8)⋊13D5, C4.41(D4×D5), C102(C4○D8), (C4×D5).66D4, C20.78(C2×D4), C4.3(C23×D5), D10.21(C2×D4), (C2×C8).245D10, (C5×D8)⋊10C22, (C8×D5)⋊14C22, D4.D58C22, (C5×D4).1C23, D4.1(C22×D5), C8.38(C22×D5), (C2×Dic20)⋊19C2, (C2×D4).180D10, D42D56C22, (C2×C40).97C22, C52C8.20C23, (C4×D5).60C23, (C22×D5).92D4, C22.137(D4×D5), (C2×C20).520C23, Dic5.123(C2×D4), (C2×Dic5).283D4, C10.104(C22×D4), (D4×C10).162C22, (C2×Dic10).201C22, (D5×C2×C8)⋊5C2, C52(C2×C4○D8), C2.77(C2×D4×D5), (C2×D4.D5)⋊26C2, (C2×D42D5)⋊24C2, (C2×C10).393(C2×D4), (C2×C4×D5).326C22, (C2×C4).610(C22×D5), (C2×C52C8).291C22, SmallGroup(320,1428)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D83D5
C1C5C10C20C4×D5C2×C4×D5C2×D42D5 — C2×D83D5
C5C10C20 — C2×D83D5
C1C22C2×C4C2×D8

Generators and relations for C2×D83D5
 G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 926 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×4], D4 [×10], Q8 [×6], C23 [×3], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×12], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×8], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10 [×4], Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5, C22×C10 [×2], C2×C4○D8, C8×D5 [×4], Dic20 [×4], C2×C52C8, D4.D5 [×8], C2×C40, C5×D8 [×4], C2×Dic10 [×2], C2×C4×D5, D42D5 [×8], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10 [×2], D5×C2×C8, C2×Dic20, D83D5 [×8], C2×D4.D5 [×2], C10×D8, C2×D42D5 [×2], C2×D83D5
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C22×D5 [×7], C2×C4○D8, D4×D5 [×2], C23×D5, D83D5 [×2], C2×D4×D5, C2×D83D5

Smallest permutation representation of C2×D83D5
On 160 points
Generators in S160
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 113)(17 157)(18 158)(19 159)(20 160)(21 153)(22 154)(23 155)(24 156)(25 143)(26 144)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 150)(34 151)(35 152)(36 145)(37 146)(38 147)(39 148)(40 149)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(57 110)(58 111)(59 112)(60 105)(61 106)(62 107)(63 108)(64 109)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 129)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 121)(80 122)
(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)
(1 60)(2 59)(3 58)(4 57)(5 64)(6 63)(7 62)(8 61)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 138)(18 137)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 160)(26 159)(27 158)(28 157)(29 156)(30 155)(31 154)(32 153)(41 74)(42 73)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 72)(81 123)(82 122)(83 121)(84 128)(85 127)(86 126)(87 125)(88 124)(89 105)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 132)(98 131)(99 130)(100 129)(101 136)(102 135)(103 134)(104 133)(113 149)(114 148)(115 147)(116 146)(117 145)(118 152)(119 151)(120 150)
(1 115 28 127 131)(2 116 29 128 132)(3 117 30 121 133)(4 118 31 122 134)(5 119 32 123 135)(6 120 25 124 136)(7 113 26 125 129)(8 114 27 126 130)(9 137 76 65 96)(10 138 77 66 89)(11 139 78 67 90)(12 140 79 68 91)(13 141 80 69 92)(14 142 73 70 93)(15 143 74 71 94)(16 144 75 72 95)(17 46 54 105 38)(18 47 55 106 39)(19 48 56 107 40)(20 41 49 108 33)(21 42 50 109 34)(22 43 51 110 35)(23 44 52 111 36)(24 45 53 112 37)(57 152 154 82 103)(58 145 155 83 104)(59 146 156 84 97)(60 147 157 85 98)(61 148 158 86 99)(62 149 159 87 100)(63 150 160 88 101)(64 151 153 81 102)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 129)(8 130)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 21)(18 22)(19 23)(20 24)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 97)(64 98)(65 96)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 145)(88 146)(113 125)(114 126)(115 127)(116 128)(117 121)(118 122)(119 123)(120 124)(153 157)(154 158)(155 159)(156 160)

G:=sub<Sym(160)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,113)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,143)(26,144)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,150)(34,151)(35,152)(36,145)(37,146)(38,147)(39,148)(40,149)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,110)(58,111)(59,112)(60,105)(61,106)(62,107)(63,108)(64,109)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,129)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,121)(80,122), (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), (1,60)(2,59)(3,58)(4,57)(5,64)(6,63)(7,62)(8,61)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,138)(18,137)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(41,74)(42,73)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,72)(81,123)(82,122)(83,121)(84,128)(85,127)(86,126)(87,125)(88,124)(89,105)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,132)(98,131)(99,130)(100,129)(101,136)(102,135)(103,134)(104,133)(113,149)(114,148)(115,147)(116,146)(117,145)(118,152)(119,151)(120,150), (1,115,28,127,131)(2,116,29,128,132)(3,117,30,121,133)(4,118,31,122,134)(5,119,32,123,135)(6,120,25,124,136)(7,113,26,125,129)(8,114,27,126,130)(9,137,76,65,96)(10,138,77,66,89)(11,139,78,67,90)(12,140,79,68,91)(13,141,80,69,92)(14,142,73,70,93)(15,143,74,71,94)(16,144,75,72,95)(17,46,54,105,38)(18,47,55,106,39)(19,48,56,107,40)(20,41,49,108,33)(21,42,50,109,34)(22,43,51,110,35)(23,44,52,111,36)(24,45,53,112,37)(57,152,154,82,103)(58,145,155,83,104)(59,146,156,84,97)(60,147,157,85,98)(61,148,158,86,99)(62,149,159,87,100)(63,150,160,88,101)(64,151,153,81,102), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,129)(8,130)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,21)(18,22)(19,23)(20,24)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,97)(64,98)(65,96)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,145)(88,146)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124)(153,157)(154,158)(155,159)(156,160)>;

G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,113)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,143)(26,144)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,150)(34,151)(35,152)(36,145)(37,146)(38,147)(39,148)(40,149)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,110)(58,111)(59,112)(60,105)(61,106)(62,107)(63,108)(64,109)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,129)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,121)(80,122), (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), (1,60)(2,59)(3,58)(4,57)(5,64)(6,63)(7,62)(8,61)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,138)(18,137)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(41,74)(42,73)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,72)(81,123)(82,122)(83,121)(84,128)(85,127)(86,126)(87,125)(88,124)(89,105)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,132)(98,131)(99,130)(100,129)(101,136)(102,135)(103,134)(104,133)(113,149)(114,148)(115,147)(116,146)(117,145)(118,152)(119,151)(120,150), (1,115,28,127,131)(2,116,29,128,132)(3,117,30,121,133)(4,118,31,122,134)(5,119,32,123,135)(6,120,25,124,136)(7,113,26,125,129)(8,114,27,126,130)(9,137,76,65,96)(10,138,77,66,89)(11,139,78,67,90)(12,140,79,68,91)(13,141,80,69,92)(14,142,73,70,93)(15,143,74,71,94)(16,144,75,72,95)(17,46,54,105,38)(18,47,55,106,39)(19,48,56,107,40)(20,41,49,108,33)(21,42,50,109,34)(22,43,51,110,35)(23,44,52,111,36)(24,45,53,112,37)(57,152,154,82,103)(58,145,155,83,104)(59,146,156,84,97)(60,147,157,85,98)(61,148,158,86,99)(62,149,159,87,100)(63,150,160,88,101)(64,151,153,81,102), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,129)(8,130)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,21)(18,22)(19,23)(20,24)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,97)(64,98)(65,96)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,145)(88,146)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124)(153,157)(154,158)(155,159)(156,160) );

G=PermutationGroup([(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,113),(17,157),(18,158),(19,159),(20,160),(21,153),(22,154),(23,155),(24,156),(25,143),(26,144),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,150),(34,151),(35,152),(36,145),(37,146),(38,147),(39,148),(40,149),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,101),(50,102),(51,103),(52,104),(53,97),(54,98),(55,99),(56,100),(57,110),(58,111),(59,112),(60,105),(61,106),(62,107),(63,108),(64,109),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,129),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,121),(80,122)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,64),(6,63),(7,62),(8,61),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,40),(17,138),(18,137),(19,144),(20,143),(21,142),(22,141),(23,140),(24,139),(25,160),(26,159),(27,158),(28,157),(29,156),(30,155),(31,154),(32,153),(41,74),(42,73),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,72),(81,123),(82,122),(83,121),(84,128),(85,127),(86,126),(87,125),(88,124),(89,105),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,132),(98,131),(99,130),(100,129),(101,136),(102,135),(103,134),(104,133),(113,149),(114,148),(115,147),(116,146),(117,145),(118,152),(119,151),(120,150)], [(1,115,28,127,131),(2,116,29,128,132),(3,117,30,121,133),(4,118,31,122,134),(5,119,32,123,135),(6,120,25,124,136),(7,113,26,125,129),(8,114,27,126,130),(9,137,76,65,96),(10,138,77,66,89),(11,139,78,67,90),(12,140,79,68,91),(13,141,80,69,92),(14,142,73,70,93),(15,143,74,71,94),(16,144,75,72,95),(17,46,54,105,38),(18,47,55,106,39),(19,48,56,107,40),(20,41,49,108,33),(21,42,50,109,34),(22,43,51,110,35),(23,44,52,111,36),(24,45,53,112,37),(57,152,154,82,103),(58,145,155,83,104),(59,146,156,84,97),(60,147,157,85,98),(61,148,158,86,99),(62,149,159,87,100),(63,150,160,88,101),(64,151,153,81,102)], [(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,129),(8,130),(9,76),(10,77),(11,78),(12,79),(13,80),(14,73),(15,74),(16,75),(17,21),(18,22),(19,23),(20,24),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44),(49,112),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,97),(64,98),(65,96),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,145),(88,146),(113,125),(114,126),(115,127),(116,128),(117,121),(118,122),(119,123),(120,124),(153,157),(154,158),(155,159),(156,160)])

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F10G···10N20A20B20C20D40A···40H
order12222222224444444444558888888810···1010···102020202040···40
size11114444101022555520202020222222101010102···28···844444···4

56 irreducible representations

dim111111122222222444
type++++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D5D10D10D10C4○D8D4×D5D4×D5D83D5
kernelC2×D83D5D5×C2×C8C2×Dic20D83D5C2×D4.D5C10×D8C2×D42D5C4×D5C2×Dic5C22×D5C2×D8C2×C8D8C2×D4C10C4C22C2
# reps111821221122848228

Matrix representation of C2×D83D5 in GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
03000
001400
000400
000040
,
400000
003200
09000
000400
000040
,
10000
01000
00100
000740
000840
,
400000
040000
00100
00006
00070

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,0,0,0,0,0,14,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,8,0,0,0,40,40],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,6,0] >;

C2×D83D5 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes_3D_5
% in TeX

G:=Group("C2xD8:3D5");
// GroupNames label

G:=SmallGroup(320,1428);
// by ID

G=gap.SmallGroup(320,1428);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,1123,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

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