direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.34D4, (C22×C4)⋊7C20, (C22×C20)⋊24C4, (C23×C4).3C10, (C23×C20).6C2, C23.34(C5×D4), C23.26(C2×C20), C24.25(C2×C10), C22.31(D4×C10), C2.C42⋊2C10, (C22×C10).154D4, C23.54(C22×C10), C22.30(C22×C20), (C23×C10).85C22, C10.73(C42⋊C2), (C22×C20).490C22, (C22×C10).445C23, C10.85(C22.D4), (C2×C4).56(C2×C20), C2.6(C10×C22⋊C4), (C2×C20).457(C2×C4), (C2×C10).598(C2×D4), (C2×C22⋊C4).4C10, (C22×C4).3(C2×C10), C2.6(C5×C42⋊C2), C22.16(C5×C4○D4), (C10×C22⋊C4).10C2, C10.134(C2×C22⋊C4), (C5×C2.C42)⋊4C2, C22.16(C5×C22⋊C4), (C2×C10).206(C4○D4), C2.1(C5×C22.D4), (C22×C10).180(C2×C4), (C2×C10).318(C22×C4), (C2×C10).143(C22⋊C4), SmallGroup(320,882)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.34D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >
Subgroups: 354 in 218 conjugacy classes, 98 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×24], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×10], C22×C4 [×4], C24, C20 [×8], C2×C10, C2×C10 [×10], C2×C10 [×12], C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C2×C20 [×4], C2×C20 [×24], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.34D4, C5×C22⋊C4 [×4], C22×C20 [×10], C22×C20 [×4], C23×C10, C5×C2.C42 [×4], C10×C22⋊C4 [×2], C23×C20, C5×C23.34D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×C20 [×6], C5×D4 [×4], C22×C10, C23.34D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C5×C4○D4 [×4], C10×C22⋊C4, C5×C42⋊C2 [×2], C5×C22.D4 [×4], C5×C23.34D4
►On 160 points
Generators in S160
(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 66)(2 67)(3 68)(4 69)(5 70)(6 152)(7 153)(8 154)(9 155)(10 151)(11 41)(12 42)(13 43)(14 44)(15 45)(16 144)(17 145)(18 141)(19 142)(20 143)(21 160)(22 156)(23 157)(24 158)(25 159)(26 39)(27 40)(28 36)(29 37)(30 38)(31 136)(32 137)(33 138)(34 139)(35 140)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 126)(97 127)(98 128)(99 129)(100 130)(101 148)(102 149)(103 150)(104 146)(105 147)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 143)(7 144)(8 145)(9 141)(10 142)(16 153)(17 154)(18 155)(19 151)(20 152)(21 140)(22 136)(23 137)(24 138)(25 139)(26 51)(27 52)(28 53)(29 54)(30 55)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(41 70)(42 66)(43 67)(44 68)(45 69)(56 93)(57 94)(58 95)(59 91)(60 92)(61 83)(62 84)(63 85)(64 81)(65 82)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 133)(97 134)(98 135)(99 131)(100 132)(101 123)(102 124)(103 125)(104 121)(105 122)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 156)(7 157)(8 158)(9 159)(10 160)(11 46)(12 47)(13 48)(14 49)(15 50)(16 137)(17 138)(18 139)(19 140)(20 136)(21 151)(22 152)(23 153)(24 154)(25 155)(26 68)(27 69)(28 70)(29 66)(30 67)(31 143)(32 144)(33 145)(34 141)(35 142)(41 53)(42 54)(43 55)(44 51)(45 52)(56 110)(57 106)(58 107)(59 108)(60 109)(61 89)(62 90)(63 86)(64 87)(65 88)(71 92)(72 93)(73 94)(74 95)(75 91)(76 85)(77 81)(78 82)(79 83)(80 84)(96 150)(97 146)(98 147)(99 148)(100 149)(101 129)(102 130)(103 126)(104 127)(105 128)(111 132)(112 133)(113 134)(114 135)(115 131)(116 125)(117 121)(118 122)(119 123)(120 124)
(1 97 29 121)(2 98 30 122)(3 99 26 123)(4 100 27 124)(5 96 28 125)(6 85 136 56)(7 81 137 57)(8 82 138 58)(9 83 139 59)(10 84 140 60)(11 133 53 103)(12 134 54 104)(13 135 55 105)(14 131 51 101)(15 132 52 102)(16 106 157 77)(17 107 158 78)(18 108 159 79)(19 109 160 80)(20 110 156 76)(21 92 142 62)(22 93 143 63)(23 94 144 64)(24 95 145 65)(25 91 141 61)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 126 46 112)(42 127 47 113)(43 128 48 114)(44 129 49 115)(45 130 50 111)
(1 32 47 7)(2 33 48 8)(3 34 49 9)(4 35 50 10)(5 31 46 6)(11 156 36 143)(12 157 37 144)(13 158 38 145)(14 159 39 141)(15 160 40 142)(16 66 23 54)(17 67 24 55)(18 68 25 51)(19 69 21 52)(20 70 22 53)(26 155 44 139)(27 151 45 140)(28 152 41 136)(29 153 42 137)(30 154 43 138)(56 150 72 133)(57 146 73 134)(58 147 74 135)(59 148 75 131)(60 149 71 132)(61 129 79 123)(62 130 80 124)(63 126 76 125)(64 127 77 121)(65 128 78 122)(81 117 87 104)(82 118 88 105)(83 119 89 101)(84 120 90 102)(85 116 86 103)(91 115 108 99)(92 111 109 100)(93 112 110 96)(94 113 106 97)(95 114 107 98)
G:=sub<Sym(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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,152)(7,153)(8,154)(9,155)(10,151)(11,41)(12,42)(13,43)(14,44)(15,45)(16,144)(17,145)(18,141)(19,142)(20,143)(21,160)(22,156)(23,157)(24,158)(25,159)(26,39)(27,40)(28,36)(29,37)(30,38)(31,136)(32,137)(33,138)(34,139)(35,140)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,126)(97,127)(98,128)(99,129)(100,130)(101,148)(102,149)(103,150)(104,146)(105,147)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132), (1,12)(2,13)(3,14)(4,15)(5,11)(6,143)(7,144)(8,145)(9,141)(10,142)(16,153)(17,154)(18,155)(19,151)(20,152)(21,140)(22,136)(23,137)(24,138)(25,139)(26,51)(27,52)(28,53)(29,54)(30,55)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(41,70)(42,66)(43,67)(44,68)(45,69)(56,93)(57,94)(58,95)(59,91)(60,92)(61,83)(62,84)(63,85)(64,81)(65,82)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,133)(97,134)(98,135)(99,131)(100,132)(101,123)(102,124)(103,125)(104,121)(105,122)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,37)(2,38)(3,39)(4,40)(5,36)(6,156)(7,157)(8,158)(9,159)(10,160)(11,46)(12,47)(13,48)(14,49)(15,50)(16,137)(17,138)(18,139)(19,140)(20,136)(21,151)(22,152)(23,153)(24,154)(25,155)(26,68)(27,69)(28,70)(29,66)(30,67)(31,143)(32,144)(33,145)(34,141)(35,142)(41,53)(42,54)(43,55)(44,51)(45,52)(56,110)(57,106)(58,107)(59,108)(60,109)(61,89)(62,90)(63,86)(64,87)(65,88)(71,92)(72,93)(73,94)(74,95)(75,91)(76,85)(77,81)(78,82)(79,83)(80,84)(96,150)(97,146)(98,147)(99,148)(100,149)(101,129)(102,130)(103,126)(104,127)(105,128)(111,132)(112,133)(113,134)(114,135)(115,131)(116,125)(117,121)(118,122)(119,123)(120,124), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111), (1,32,47,7)(2,33,48,8)(3,34,49,9)(4,35,50,10)(5,31,46,6)(11,156,36,143)(12,157,37,144)(13,158,38,145)(14,159,39,141)(15,160,40,142)(16,66,23,54)(17,67,24,55)(18,68,25,51)(19,69,21,52)(20,70,22,53)(26,155,44,139)(27,151,45,140)(28,152,41,136)(29,153,42,137)(30,154,43,138)(56,150,72,133)(57,146,73,134)(58,147,74,135)(59,148,75,131)(60,149,71,132)(61,129,79,123)(62,130,80,124)(63,126,76,125)(64,127,77,121)(65,128,78,122)(81,117,87,104)(82,118,88,105)(83,119,89,101)(84,120,90,102)(85,116,86,103)(91,115,108,99)(92,111,109,100)(93,112,110,96)(94,113,106,97)(95,114,107,98)>;
G:=Group( (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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,152)(7,153)(8,154)(9,155)(10,151)(11,41)(12,42)(13,43)(14,44)(15,45)(16,144)(17,145)(18,141)(19,142)(20,143)(21,160)(22,156)(23,157)(24,158)(25,159)(26,39)(27,40)(28,36)(29,37)(30,38)(31,136)(32,137)(33,138)(34,139)(35,140)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,126)(97,127)(98,128)(99,129)(100,130)(101,148)(102,149)(103,150)(104,146)(105,147)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132), (1,12)(2,13)(3,14)(4,15)(5,11)(6,143)(7,144)(8,145)(9,141)(10,142)(16,153)(17,154)(18,155)(19,151)(20,152)(21,140)(22,136)(23,137)(24,138)(25,139)(26,51)(27,52)(28,53)(29,54)(30,55)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(41,70)(42,66)(43,67)(44,68)(45,69)(56,93)(57,94)(58,95)(59,91)(60,92)(61,83)(62,84)(63,85)(64,81)(65,82)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,133)(97,134)(98,135)(99,131)(100,132)(101,123)(102,124)(103,125)(104,121)(105,122)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,37)(2,38)(3,39)(4,40)(5,36)(6,156)(7,157)(8,158)(9,159)(10,160)(11,46)(12,47)(13,48)(14,49)(15,50)(16,137)(17,138)(18,139)(19,140)(20,136)(21,151)(22,152)(23,153)(24,154)(25,155)(26,68)(27,69)(28,70)(29,66)(30,67)(31,143)(32,144)(33,145)(34,141)(35,142)(41,53)(42,54)(43,55)(44,51)(45,52)(56,110)(57,106)(58,107)(59,108)(60,109)(61,89)(62,90)(63,86)(64,87)(65,88)(71,92)(72,93)(73,94)(74,95)(75,91)(76,85)(77,81)(78,82)(79,83)(80,84)(96,150)(97,146)(98,147)(99,148)(100,149)(101,129)(102,130)(103,126)(104,127)(105,128)(111,132)(112,133)(113,134)(114,135)(115,131)(116,125)(117,121)(118,122)(119,123)(120,124), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111), (1,32,47,7)(2,33,48,8)(3,34,49,9)(4,35,50,10)(5,31,46,6)(11,156,36,143)(12,157,37,144)(13,158,38,145)(14,159,39,141)(15,160,40,142)(16,66,23,54)(17,67,24,55)(18,68,25,51)(19,69,21,52)(20,70,22,53)(26,155,44,139)(27,151,45,140)(28,152,41,136)(29,153,42,137)(30,154,43,138)(56,150,72,133)(57,146,73,134)(58,147,74,135)(59,148,75,131)(60,149,71,132)(61,129,79,123)(62,130,80,124)(63,126,76,125)(64,127,77,121)(65,128,78,122)(81,117,87,104)(82,118,88,105)(83,119,89,101)(84,120,90,102)(85,116,86,103)(91,115,108,99)(92,111,109,100)(93,112,110,96)(94,113,106,97)(95,114,107,98) );
G=PermutationGroup([(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)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,152),(7,153),(8,154),(9,155),(10,151),(11,41),(12,42),(13,43),(14,44),(15,45),(16,144),(17,145),(18,141),(19,142),(20,143),(21,160),(22,156),(23,157),(24,158),(25,159),(26,39),(27,40),(28,36),(29,37),(30,38),(31,136),(32,137),(33,138),(34,139),(35,140),(46,53),(47,54),(48,55),(49,51),(50,52),(56,76),(57,77),(58,78),(59,79),(60,80),(61,75),(62,71),(63,72),(64,73),(65,74),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,126),(97,127),(98,128),(99,129),(100,130),(101,148),(102,149),(103,150),(104,146),(105,147),(111,124),(112,125),(113,121),(114,122),(115,123),(116,133),(117,134),(118,135),(119,131),(120,132)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,143),(7,144),(8,145),(9,141),(10,142),(16,153),(17,154),(18,155),(19,151),(20,152),(21,140),(22,136),(23,137),(24,138),(25,139),(26,51),(27,52),(28,53),(29,54),(30,55),(31,156),(32,157),(33,158),(34,159),(35,160),(36,46),(37,47),(38,48),(39,49),(40,50),(41,70),(42,66),(43,67),(44,68),(45,69),(56,93),(57,94),(58,95),(59,91),(60,92),(61,83),(62,84),(63,85),(64,81),(65,82),(71,109),(72,110),(73,106),(74,107),(75,108),(76,86),(77,87),(78,88),(79,89),(80,90),(96,133),(97,134),(98,135),(99,131),(100,132),(101,123),(102,124),(103,125),(104,121),(105,122),(111,149),(112,150),(113,146),(114,147),(115,148),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,37),(2,38),(3,39),(4,40),(5,36),(6,156),(7,157),(8,158),(9,159),(10,160),(11,46),(12,47),(13,48),(14,49),(15,50),(16,137),(17,138),(18,139),(19,140),(20,136),(21,151),(22,152),(23,153),(24,154),(25,155),(26,68),(27,69),(28,70),(29,66),(30,67),(31,143),(32,144),(33,145),(34,141),(35,142),(41,53),(42,54),(43,55),(44,51),(45,52),(56,110),(57,106),(58,107),(59,108),(60,109),(61,89),(62,90),(63,86),(64,87),(65,88),(71,92),(72,93),(73,94),(74,95),(75,91),(76,85),(77,81),(78,82),(79,83),(80,84),(96,150),(97,146),(98,147),(99,148),(100,149),(101,129),(102,130),(103,126),(104,127),(105,128),(111,132),(112,133),(113,134),(114,135),(115,131),(116,125),(117,121),(118,122),(119,123),(120,124)], [(1,97,29,121),(2,98,30,122),(3,99,26,123),(4,100,27,124),(5,96,28,125),(6,85,136,56),(7,81,137,57),(8,82,138,58),(9,83,139,59),(10,84,140,60),(11,133,53,103),(12,134,54,104),(13,135,55,105),(14,131,51,101),(15,132,52,102),(16,106,157,77),(17,107,158,78),(18,108,159,79),(19,109,160,80),(20,110,156,76),(21,92,142,62),(22,93,143,63),(23,94,144,64),(24,95,145,65),(25,91,141,61),(31,86,152,72),(32,87,153,73),(33,88,154,74),(34,89,155,75),(35,90,151,71),(36,150,70,116),(37,146,66,117),(38,147,67,118),(39,148,68,119),(40,149,69,120),(41,126,46,112),(42,127,47,113),(43,128,48,114),(44,129,49,115),(45,130,50,111)], [(1,32,47,7),(2,33,48,8),(3,34,49,9),(4,35,50,10),(5,31,46,6),(11,156,36,143),(12,157,37,144),(13,158,38,145),(14,159,39,141),(15,160,40,142),(16,66,23,54),(17,67,24,55),(18,68,25,51),(19,69,21,52),(20,70,22,53),(26,155,44,139),(27,151,45,140),(28,152,41,136),(29,153,42,137),(30,154,43,138),(56,150,72,133),(57,146,73,134),(58,147,74,135),(59,148,75,131),(60,149,71,132),(61,129,79,123),(62,130,80,124),(63,126,76,125),(64,127,77,121),(65,128,78,122),(81,117,87,104),(82,118,88,105),(83,119,89,101),(84,120,90,102),(85,116,86,103),(91,115,108,99),(92,111,109,100),(93,112,110,96),(94,113,106,97),(95,114,107,98)])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C23.34D4 | C5×C2.C42 | C10×C22⋊C4 | C23×C20 | C22×C20 | C23.34D4 | C2.C42 | C2×C22⋊C4 | C23×C4 | C22×C4 | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 4 | 8 | 16 | 32 |
Matrix representation of C5×C23.34D4 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 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 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[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],[9,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,1,0],[32,0,0,0,0,0,0,9,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,32,0] >;
C5×C23.34D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{34}D_4 % in TeX
G:=Group("C5xC2^3.34D4"); // GroupNames label
G:=SmallGroup(320,882);
// by ID
G=gap.SmallGroup(320,882);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,226]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations