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## G = C2×D8⋊3D5order 320 = 26·5

### Direct product of C2 and D8⋊3D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D8⋊3D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4⋊2D5 — C2×D8⋊3D5
 Lower central C5 — C10 — C20 — C2×D8⋊3D5
 Upper central C1 — C22 — C2×C4 — C2×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 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 4 4 10 10 2 2 5 5 5 5 20 20 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C4○D8 D4×D5 D4×D5 D8⋊3D5 kernel C2×D8⋊3D5 D5×C2×C8 C2×Dic20 D8⋊3D5 C2×D4.D5 C10×D8 C2×D4⋊2D5 C4×D5 C2×Dic5 C22×D5 C2×D8 C2×C8 D8 C2×D4 C10 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 2 2 8 4 8 2 2 8

Matrix representation of C2×D83D5 in GL5(𝔽41)

 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 32 0 0 0 9 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 40 0 0 0 8 40
,
 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 7 0

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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