Copied to
clipboard

G = D5×C2.D8order 320 = 26·5

Direct product of D5 and C2.D8

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

Aliases: D5×C2.D8, D10.23D8, D10.12Q16, (C8×D5)⋊1C4, C813(C4×D5), C2.4(D5×D8), C4015(C2×C4), C2.4(D5×Q16), C4.29(Q8×D5), C405C420C2, C10.28(C2×D8), (C4×D5).15Q8, C20.20(C2×Q8), C4⋊C4.169D10, (C2×C8).229D10, C10.23(C2×Q16), C22.89(D4×D5), D10.37(C4⋊C4), C10.D819C2, (C2×C40).81C22, Dic5.17(C4⋊C4), (C2×C20).295C23, C20.107(C22×C4), (C2×Dic5).147D4, (C22×D5).155D4, C4⋊Dic5.121C22, C52(C2×C2.D8), (D5×C2×C8).2C2, C4.79(C2×C4×D5), (D5×C4⋊C4).7C2, C2.14(D5×C4⋊C4), (C5×C2.D8)⋊3C2, C52C828(C2×C4), C10.36(C2×C4⋊C4), (C4×D5).75(C2×C4), (C2×C10).300(C2×D4), (C5×C4⋊C4).88C22, (C2×C4×D5).304C22, (C2×C4).398(C22×D5), (C2×C52C8).241C22, SmallGroup(320,506)

Series: Derived Chief Lower central Upper central

C1C20 — D5×C2.D8
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — D5×C2.D8
C5C10C20 — D5×C2.D8
C1C22C2×C4C2.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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444444444558888888810···102020202020···2040···40
size11115555224444101020202020222222101010102···244448···84···4

56 irreducible representations

dim11111112222222224444
type++++++-++++-++-++-
imageC1C2C2C2C2C2C4Q8D4D4D5D8Q16D10D10C4×D5Q8×D5D4×D5D5×D8D5×Q16
kernelD5×C2.D8C10.D8C405C4C5×C2.D8D5×C4⋊C4D5×C2×C8C8×D5C4×D5C2×Dic5C22×D5C2.D8D10D10C4⋊C4C2×C8C8C4C22C2C2
# reps12112182112444282244

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

10000
040100
033700
00010
00001
,
400000
040000
033100
000400
000040
,
400000
01000
00100
000400
000040
,
10000
040000
004000
0002912
0002929
,
320000
040000
004000
0001434
0003427

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