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## G = C2×D5×Q16order 320 = 26·5

### Direct product of C2, D5 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D5×Q16
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×Q8×D5 — C2×D5×Q16
 Lower central C5 — C10 — C20 — C2×D5×Q16
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×D5×Q16
G = < a,b,c,d,e | a2=b5=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 862 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×17], Q8 [×4], Q8 [×16], C23, D5 [×4], C10, C10 [×2], C2×C8, C2×C8 [×5], Q16 [×4], Q16 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×16], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C22×C8, C2×Q16, C2×Q16 [×11], C22×Q8 [×2], C52C8 [×2], C40 [×2], Dic10 [×4], Dic10 [×10], C4×D5 [×4], C4×D5 [×8], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C22×D5, C22×Q16, C8×D5 [×4], Dic20 [×4], C2×C52C8, C5⋊Q16 [×8], C2×C40, C5×Q16 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×8], Q8×D5 [×4], Q8×C10 [×2], D5×C2×C8, C2×Dic20, D5×Q16 [×8], C2×C5⋊Q16 [×2], C10×Q16, C2×Q8×D5 [×2], C2×D5×Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, C22×D5 [×7], C22×Q16, D4×D5 [×2], C23×D5, D5×Q16 [×2], C2×D4×D5, C2×D5×Q16

Smallest permutation representation of C2×D5×Q16
On 160 points
Generators in S160
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 97)(24 98)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 145)(48 146)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(81 124)(82 125)(83 126)(84 127)(85 128)(86 121)(87 122)(88 123)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)
(1 138 15 68 152)(2 139 16 69 145)(3 140 9 70 146)(4 141 10 71 147)(5 142 11 72 148)(6 143 12 65 149)(7 144 13 66 150)(8 137 14 67 151)(17 80 159 96 84)(18 73 160 89 85)(19 74 153 90 86)(20 75 154 91 87)(21 76 155 92 88)(22 77 156 93 81)(23 78 157 94 82)(24 79 158 95 83)(25 58 131 122 102)(26 59 132 123 103)(27 60 133 124 104)(28 61 134 125 97)(29 62 135 126 98)(30 63 136 127 99)(31 64 129 128 100)(32 57 130 121 101)(33 51 111 115 45)(34 52 112 116 46)(35 53 105 117 47)(36 54 106 118 48)(37 55 107 119 41)(38 56 108 120 42)(39 49 109 113 43)(40 50 110 114 44)
(1 152)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(17 96)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(65 143)(66 144)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 160)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)
(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 58 5 62)(2 57 6 61)(3 64 7 60)(4 63 8 59)(9 128 13 124)(10 127 14 123)(11 126 15 122)(12 125 16 121)(17 115 21 119)(18 114 22 118)(19 113 23 117)(20 120 24 116)(25 148 29 152)(26 147 30 151)(27 146 31 150)(28 145 32 149)(33 155 37 159)(34 154 38 158)(35 153 39 157)(36 160 40 156)(41 80 45 76)(42 79 46 75)(43 78 47 74)(44 77 48 73)(49 94 53 90)(50 93 54 89)(51 92 55 96)(52 91 56 95)(65 97 69 101)(66 104 70 100)(67 103 71 99)(68 102 72 98)(81 106 85 110)(82 105 86 109)(83 112 87 108)(84 111 88 107)(129 144 133 140)(130 143 134 139)(131 142 135 138)(132 141 136 137)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 5 5 5 5 2 2 4 4 4 4 10 10 20 20 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 8 ··· 8 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 Q16 D10 D10 D10 D4×D5 D4×D5 D5×Q16 kernel C2×D5×Q16 D5×C2×C8 C2×Dic20 D5×Q16 C2×C5⋊Q16 C10×Q16 C2×Q8×D5 C4×D5 C2×Dic5 C22×D5 C2×Q16 D10 C2×C8 Q16 C2×Q8 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 2 8 2 8 4 2 2 8

Matrix representation of C2×D5×Q16 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 40 1 0 0 33 7
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 33 1
,
 24 35 0 0 7 0 0 0 0 0 40 0 0 0 0 40
,
 37 40 0 0 17 4 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,0,1],[24,7,0,0,35,0,0,0,0,0,40,0,0,0,0,40],[37,17,0,0,40,4,0,0,0,0,40,0,0,0,0,40] >;

C2×D5×Q16 in GAP, Magma, Sage, TeX

C_2\times D_5\times Q_{16}
% in TeX

G:=Group("C2xD5xQ16");
// GroupNames label

G:=SmallGroup(320,1435);
// by ID

G=gap.SmallGroup(320,1435);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,185,136,438,235,102,12550]);
// Polycyclic

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

׿
×
𝔽