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G = C2×Q8.D10order 320 = 26·5

Direct product of C2 and Q8.D10

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

Aliases: C2×Q8.D10, Q1611D10, D4017C22, C40.34C23, C20.12C24, D20.7C23, C4.48(D4×D5), (C2×D40)⋊20C2, C104(C4○D8), (C2×Q16)⋊13D5, (C10×Q16)⋊8C2, (C4×D5).70D4, C20.87(C2×D4), D10.23(C2×D4), (C2×C8).247D10, (C8×D5)⋊15C22, Q8⋊D510C22, (C5×Q16)⋊9C22, C4.12(C23×D5), C8.40(C22×D5), (C5×Q8).6C23, Q8.6(C22×D5), (C2×C40).99C22, C52C8.23C23, (C2×Q8).154D10, Q82D57C22, (C22×D5).94D4, (C4×D5).64C23, C22.144(D4×D5), (C2×C20).529C23, (C2×Dic5).285D4, Dic5.125(C2×D4), C10.113(C22×D4), (C2×D20).186C22, (Q8×C10).151C22, (D5×C2×C8)⋊6C2, C54(C2×C4○D8), C2.86(C2×D4×D5), (C2×Q8⋊D5)⋊28C2, (C2×Q82D5)⋊16C2, (C2×C10).402(C2×D4), (C2×C4×D5).330C22, (C2×C4).617(C22×D5), (C2×C52C8).294C22, SmallGroup(320,1437)

Series: Derived Chief Lower central Upper central

C1C20 — C2×Q8.D10
C1C5C10C20C4×D5C2×C4×D5C2×Q82D5 — C2×Q8.D10
C5C10C20 — C2×Q8.D10
C1C22C2×C4C2×Q16

Generators and relations for C2×Q8.D10
 G = < a,b,c,d,e | a2=b4=e2=1, c2=d10=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d9 >

Subgroups: 1054 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 [×14], Q8 [×4], Q8 [×2], C23 [×3], D5 [×6], C10, C10 [×2], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×10], C2×C10, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×8], D20 [×4], D20 [×10], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C2×C4○D8, C8×D5 [×4], D40 [×4], C2×C52C8, Q8⋊D5 [×8], C2×C40, C5×Q16 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×2], Q82D5 [×8], Q82D5 [×4], Q8×C10 [×2], D5×C2×C8, C2×D40, Q8.D10 [×8], C2×Q8⋊D5 [×2], C10×Q16, C2×Q82D5 [×2], C2×Q8.D10
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, Q8.D10 [×2], C2×D4×D5, C2×Q8.D10

Smallest permutation representation of C2×Q8.D10
On 160 points
Generators in S160
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 153)(120 154)
(1 67 11 77)(2 78 12 68)(3 69 13 79)(4 80 14 70)(5 71 15 61)(6 62 16 72)(7 73 17 63)(8 64 18 74)(9 75 19 65)(10 66 20 76)(21 84 31 94)(22 95 32 85)(23 86 33 96)(24 97 34 87)(25 88 35 98)(26 99 36 89)(27 90 37 100)(28 81 38 91)(29 92 39 82)(30 83 40 93)(41 146 51 156)(42 157 52 147)(43 148 53 158)(44 159 54 149)(45 150 55 160)(46 141 56 151)(47 152 57 142)(48 143 58 153)(49 154 59 144)(50 145 60 155)(101 139 111 129)(102 130 112 140)(103 121 113 131)(104 132 114 122)(105 123 115 133)(106 134 116 124)(107 125 117 135)(108 136 118 126)(109 127 119 137)(110 138 120 128)
(1 118 11 108)(2 127 12 137)(3 120 13 110)(4 129 14 139)(5 102 15 112)(6 131 16 121)(7 104 17 114)(8 133 18 123)(9 106 19 116)(10 135 20 125)(21 160 31 150)(22 46 32 56)(23 142 33 152)(24 48 34 58)(25 144 35 154)(26 50 36 60)(27 146 37 156)(28 52 38 42)(29 148 39 158)(30 54 40 44)(41 100 51 90)(43 82 53 92)(45 84 55 94)(47 86 57 96)(49 88 59 98)(61 130 71 140)(62 113 72 103)(63 132 73 122)(64 115 74 105)(65 134 75 124)(66 117 76 107)(67 136 77 126)(68 119 78 109)(69 138 79 128)(70 101 80 111)(81 157 91 147)(83 159 93 149)(85 141 95 151)(87 143 97 153)(89 145 99 155)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 137)(22 136)(23 135)(24 134)(25 133)(26 132)(27 131)(28 130)(29 129)(30 128)(31 127)(32 126)(33 125)(34 124)(35 123)(36 122)(37 121)(38 140)(39 139)(40 138)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222222224444444444558888888810···102020202020···2040···40
size11111010202020202244445555222222101010102···244448···84···4

56 irreducible representations

dim111111122222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D5D10D10D10C4○D8D4×D5D4×D5Q8.D10
kernelC2×Q8.D10D5×C2×C8C2×D40Q8.D10C2×Q8⋊D5C10×Q16C2×Q82D5C4×D5C2×Dic5C22×D5C2×Q16C2×C8Q16C2×Q8C10C4C22C2
# reps111821221122848228

Matrix representation of C2×Q8.D10 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
1000
0100
00040
0010
,
40000
04000
00320
0009
,
63500
6100
001515
001526
,
04000
40000
001229
002929
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[6,6,0,0,35,1,0,0,0,0,15,15,0,0,15,26],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29] >;

C2×Q8.D10 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_{10}
% in TeX

G:=Group("C2xQ8.D10");
// GroupNames label

G:=SmallGroup(320,1437);
// by ID

G=gap.SmallGroup(320,1437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,1123,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=b^2*d^9>;
// generators/relations

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