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G = C22×D42D5order 320 = 26·5

Direct product of C22 and D42D5

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

Aliases: C22×D42D5, C10.6C25, C20.41C24, D10.2C24, C24.60D10, Dic108C23, Dic5.25C24, (C2×D4)⋊48D10, (C5×D4)⋊7C23, D46(C22×D5), (C4×D5)⋊5C23, C5⋊D42C23, C2.7(D5×C24), (C22×D4)⋊14D5, (C2×C10).1C24, C4.41(C23×D5), (D4×C10)⋊50C22, C22.1(C23×D5), (C2×C20).563C23, (C23×Dic5)⋊11C2, (C2×Dic5)⋊11C23, (C22×C4).389D10, (C2×Dic10)⋊71C22, (C22×Dic10)⋊23C2, (C23×C10).81C22, C23.348(C22×D5), (C22×C20).299C22, (C22×C10).434C23, (C22×Dic5)⋊52C22, (C23×D5).129C22, (C22×D5).256C23, (D4×C2×C10)⋊10C2, C102(C2×C4○D4), (D5×C22×C4)⋊9C2, C52(C22×C4○D4), (C2×C4×D5)⋊59C22, (C2×C10)⋊16(C4○D4), (C22×C5⋊D4)⋊20C2, (C2×C5⋊D4)⋊51C22, (C2×C4).643(C22×D5), SmallGroup(320,1613)

Series: Derived Chief Lower central Upper central

C1C10 — C22×D42D5
C1C5C10D10C22×D5C23×D5D5×C22×C4 — C22×D42D5
C5C10 — C22×D42D5
C1C23C22×D4

Generators and relations for C22×D42D5
 G = < a,b,c,d,e,f | a2=b2=c4=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=c2d, fef=e-1 >

Subgroups: 2398 in 890 conjugacy classes, 463 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×4], C4 [×12], C22 [×15], C22 [×40], C5, C2×C4 [×6], C2×C4 [×66], D4 [×16], D4 [×32], Q8 [×16], C23, C23 [×12], C23 [×18], D5 [×4], C10, C10 [×6], C10 [×8], C22×C4, C22×C4 [×39], C2×D4 [×12], C2×D4 [×24], C2×Q8 [×12], C4○D4 [×64], C24 [×2], C24, Dic5 [×12], C20 [×4], D10 [×4], D10 [×12], C2×C10 [×15], C2×C10 [×24], C23×C4 [×3], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×24], Dic10 [×16], C4×D5 [×16], C2×Dic5 [×50], C5⋊D4 [×32], C2×C20 [×6], C5×D4 [×16], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×12], C22×C10 [×8], C22×C4○D4, C2×Dic10 [×12], C2×C4×D5 [×12], D42D5 [×64], C22×Dic5, C22×Dic5 [×26], C2×C5⋊D4 [×24], C22×C20, D4×C10 [×12], C23×D5, C23×C10 [×2], C22×Dic10, D5×C22×C4, C2×D42D5 [×24], C23×Dic5 [×2], C22×C5⋊D4 [×2], D4×C2×C10, C22×D42D5
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C4○D4 [×4], C24 [×31], D10 [×15], C2×C4○D4 [×6], C25, C22×D5 [×35], C22×C4○D4, D42D5 [×4], C23×D5 [×15], C2×D42D5 [×6], D5×C24, C22×D42D5

Smallest permutation representation of C22×D42D5
On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(81 126)(82 127)(83 128)(84 129)(85 130)(86 121)(87 122)(88 123)(89 124)(90 125)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(1 106 6 101)(2 107 7 102)(3 108 8 103)(4 109 9 104)(5 110 10 105)(11 116 16 111)(12 117 17 112)(13 118 18 113)(14 119 19 114)(15 120 20 115)(21 81 26 86)(22 82 27 87)(23 83 28 88)(24 84 29 89)(25 85 30 90)(31 91 36 96)(32 92 37 97)(33 93 38 98)(34 94 39 99)(35 95 40 100)(41 146 46 141)(42 147 47 142)(43 148 48 143)(44 149 49 144)(45 150 50 145)(51 156 56 151)(52 157 57 152)(53 158 58 153)(54 159 59 154)(55 160 60 155)(61 121 66 126)(62 122 67 127)(63 123 68 128)(64 124 69 129)(65 125 70 130)(71 131 76 136)(72 132 77 137)(73 133 78 138)(74 134 79 139)(75 135 80 140)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)
(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 85)(2 84)(3 83)(4 82)(5 81)(6 90)(7 89)(8 88)(9 87)(10 86)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 105)(22 104)(23 103)(24 102)(25 101)(26 110)(27 109)(28 108)(29 107)(30 106)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 125)(42 124)(43 123)(44 122)(45 121)(46 130)(47 129)(48 128)(49 127)(50 126)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 145)(62 144)(63 143)(64 142)(65 141)(66 150)(67 149)(68 148)(69 147)(70 146)(71 160)(72 159)(73 158)(74 157)(75 156)(76 155)(77 154)(78 153)(79 152)(80 151)

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

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

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

80 conjugacy classes

class 1 2A···2G2H···2O2P2Q2R2S4A4B4C4D4E···4L4M···4T5A5B10A···10N10O···10AD20A···20H
order12···22···2222244444···44···45510···1010···1020···20
size11···12···21010101022225···510···10222···24···44···4

80 irreducible representations

dim1111111222224
type+++++++++++-
imageC1C2C2C2C2C2C2D5C4○D4D10D10D10D42D5
kernelC22×D42D5C22×Dic10D5×C22×C4C2×D42D5C23×Dic5C22×C5⋊D4D4×C2×C10C22×D4C2×C10C22×C4C2×D4C24C22
# reps111242212822448

Matrix representation of C22×D42D5 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
10000
040000
004000
00010
00001
,
10000
01000
00100
00090
000032
,
400000
01000
00100
000032
00090
,
10000
0344000
01000
00010
00001
,
10000
0344000
07700
000400
00001

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,32],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,32,0],[1,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,7,0,0,0,40,7,0,0,0,0,0,40,0,0,0,0,0,1] >;

C22×D42D5 in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes_2D_5
% in TeX

G:=Group("C2^2xD4:2D5");
// GroupNames label

G:=SmallGroup(320,1613);
// by ID

G=gap.SmallGroup(320,1613);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,136,1684,235,12550]);
// Polycyclic

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

׿
×
𝔽