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## G = C2×D8.D5order 320 = 26·5

### Direct product of C2 and D8.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×D8.D5
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — C2×Dic20 — C2×D8.D5
 Lower central C5 — C10 — C20 — C40 — C2×D8.D5
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×D8

Generators and relations for C2×D8.D5
G = < a,b,c,d,e | a2=b8=c2=d5=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >

Subgroups: 350 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C10, C10 [×2], C10 [×2], C16 [×2], C2×C8, D8 [×2], D8, Q16 [×3], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×C16, SD32 [×4], C2×D8, C2×Q16, C40 [×2], Dic10 [×3], C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C2×SD32, C52C16 [×2], Dic20 [×2], Dic20, C2×C40, C5×D8 [×2], C5×D8, C2×Dic10, D4×C10, C2×C52C16, D8.D5 [×4], C2×Dic20, C10×D8, C2×D8.D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], SD32 [×2], C2×D8, C5⋊D4 [×2], C22×D5, C2×SD32, D4⋊D5 [×2], C2×C5⋊D4, D8.D5 [×2], C2×D4⋊D5, C2×D8.D5

Smallest permutation representation of C2×D8.D5
On 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 148)(10 149)(11 150)(12 151)(13 152)(14 145)(15 146)(16 147)(17 144)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 132)(26 133)(27 134)(28 135)(29 136)(30 129)(31 130)(32 131)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 121)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 104)(7 103)(8 102)(9 148)(10 147)(11 146)(12 145)(13 152)(14 151)(15 150)(16 149)(17 138)(18 137)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 126)(36 125)(37 124)(38 123)(39 122)(40 121)(41 74)(42 73)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 90)(58 89)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 112)(113 157)(114 156)(115 155)(116 154)(117 153)(118 160)(119 159)(120 158)
(1 54 48 66 63)(2 55 41 67 64)(3 56 42 68 57)(4 49 43 69 58)(5 50 44 70 59)(6 51 45 71 60)(7 52 46 72 61)(8 53 47 65 62)(9 155 36 18 27)(10 156 37 19 28)(11 157 38 20 29)(12 158 39 21 30)(13 159 40 22 31)(14 160 33 23 32)(15 153 34 24 25)(16 154 35 17 26)(73 108 90 99 81)(74 109 91 100 82)(75 110 92 101 83)(76 111 93 102 84)(77 112 94 103 85)(78 105 95 104 86)(79 106 96 97 87)(80 107 89 98 88)(113 123 143 132 146)(114 124 144 133 147)(115 125 137 134 148)(116 126 138 135 149)(117 127 139 136 150)(118 128 140 129 151)(119 121 141 130 152)(120 122 142 131 145)
(1 122 5 126)(2 121 6 125)(3 128 7 124)(4 127 8 123)(9 77 13 73)(10 76 14 80)(11 75 15 79)(12 74 16 78)(17 95 21 91)(18 94 22 90)(19 93 23 89)(20 92 24 96)(25 106 29 110)(26 105 30 109)(27 112 31 108)(28 111 32 107)(33 98 37 102)(34 97 38 101)(35 104 39 100)(36 103 40 99)(41 152 45 148)(42 151 46 147)(43 150 47 146)(44 149 48 145)(49 117 53 113)(50 116 54 120)(51 115 55 119)(52 114 56 118)(57 140 61 144)(58 139 62 143)(59 138 63 142)(60 137 64 141)(65 132 69 136)(66 131 70 135)(67 130 71 134)(68 129 72 133)(81 155 85 159)(82 154 86 158)(83 153 87 157)(84 160 88 156)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 16A ··· 16H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 16 ··· 16 20 20 20 20 40 ··· 40 size 1 1 1 1 8 8 2 2 40 40 2 2 2 2 2 2 2 ··· 2 8 ··· 8 10 ··· 10 4 4 4 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 SD32 C5⋊D4 C5⋊D4 D4⋊D5 D4⋊D5 D8.D5 kernel C2×D8.D5 C2×C5⋊2C16 D8.D5 C2×Dic20 C10×D8 C40 C2×C20 C2×D8 C20 C2×C10 C2×C8 D8 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 2 2 2 4 8 4 4 2 2 8

Matrix representation of C2×D8.D5 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 230 11 0 0 230 230
,
 240 0 0 0 0 240 0 0 0 0 230 11 0 0 11 11
,
 52 240 0 0 53 240 0 0 0 0 1 0 0 0 0 1
,
 239 175 0 0 11 2 0 0 0 0 103 41 0 0 41 138
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,230,11,0,0,11,11],[52,53,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[239,11,0,0,175,2,0,0,0,0,103,41,0,0,41,138] >;

C2×D8.D5 in GAP, Magma, Sage, TeX

C_2\times D_8.D_5
% in TeX

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

G:=SmallGroup(320,775);
// by ID

G=gap.SmallGroup(320,775);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,675,185,192,1684,438,102,12550]);
// Polycyclic

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

׿
×
𝔽