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## G = D5×C2.D8order 320 = 26·5

### Direct product of D5 and C2.D8

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C2.D8
G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 478 in 130 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2.D8, C2.D8 [×3], C2×C4⋊C4 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C2.D8, C8×D5 [×4], C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5 [×2], C10.D8 [×2], C405C4, C5×C2.D8, D5×C4⋊C4 [×2], D5×C2×C8, D5×C2.D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, D10 [×3], C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, C4×D5 [×2], C22×D5, C2×C2.D8, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×D8, D5×Q16, D5×C2.D8

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

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

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

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

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 2 4 4 4 4 type + + + + + + - + + + + - + + - + + - image C1 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 D8 Q16 D10 D10 C4×D5 Q8×D5 D4×D5 D5×D8 D5×Q16 kernel D5×C2.D8 C10.D8 C40⋊5C4 C5×C2.D8 D5×C4⋊C4 D5×C2×C8 C8×D5 C4×D5 C2×Dic5 C22×D5 C2.D8 D10 D10 C4⋊C4 C2×C8 C8 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 2 1 1 2 4 4 4 2 8 2 2 4 4

Matrix representation of D5×C2.D8 in GL5(𝔽41)

 1 0 0 0 0 0 40 1 0 0 0 33 7 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 0 0 0 0 33 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 29 12 0 0 0 29 29
,
 32 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 14 34 0 0 0 34 27

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,33,0,0,0,1,7,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,33,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,29,29,0,0,0,12,29],[32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,14,34,0,0,0,34,27] >;

D5×C2.D8 in GAP, Magma, Sage, TeX

D_5\times C_2.D_8
% in TeX

G:=Group("D5xC2.D8");
// GroupNames label

G:=SmallGroup(320,506);
// by ID

G=gap.SmallGroup(320,506);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=1,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,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