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, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.34D4, C5×C22⋊C4, C22×C20, C22×C20, C23×C10, C5×C2.C42, C10×C22⋊C4, C23×C20, C5×C23.34D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C4○D4, C20, C2×C10, C2×C22⋊C4, C42⋊C2, C22.D4, C2×C20, C5×D4, C22×C10, C23.34D4, C5×C22⋊C4, C22×C20, D4×C10, C5×C4○D4, C10×C22⋊C4, C5×C42⋊C2, C5×C22.D4, 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)(6 142)(7 143)(8 144)(9 145)(10 141)(11 36)(12 37)(13 38)(14 39)(15 40)(16 154)(17 155)(18 151)(19 152)(20 153)(21 160)(22 156)(23 157)(24 158)(25 159)(26 51)(27 52)(28 53)(29 54)(30 55)(31 128)(32 129)(33 130)(34 126)(35 127)(41 60)(42 56)(43 57)(44 58)(45 59)(46 79)(47 80)(48 76)(49 77)(50 78)(61 75)(62 71)(63 72)(64 73)(65 74)(81 100)(82 96)(83 97)(84 98)(85 99)(86 137)(87 138)(88 139)(89 140)(90 136)(91 108)(92 109)(93 110)(94 106)(95 107)(101 148)(102 149)(103 150)(104 146)(105 147)(111 132)(112 133)(113 134)(114 135)(115 131)(116 125)(117 121)(118 122)(119 123)(120 124)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 151)(7 152)(8 153)(9 154)(10 155)(11 45)(12 41)(13 42)(14 43)(15 44)(16 145)(17 141)(18 142)(19 143)(20 144)(21 127)(22 128)(23 129)(24 130)(25 126)(31 156)(32 157)(33 158)(34 159)(35 160)(36 59)(37 60)(38 56)(39 57)(40 58)(46 83)(47 84)(48 85)(49 81)(50 82)(51 67)(52 68)(53 69)(54 70)(55 66)(61 91)(62 92)(63 93)(64 94)(65 95)(71 109)(72 110)(73 106)(74 107)(75 108)(76 99)(77 100)(78 96)(79 97)(80 98)(86 123)(87 124)(88 125)(89 121)(90 122)(101 131)(102 132)(103 133)(104 134)(105 135)(111 149)(112 150)(113 146)(114 147)(115 148)(116 139)(117 140)(118 136)(119 137)(120 138)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 127)(7 128)(8 129)(9 130)(10 126)(11 70)(12 66)(13 67)(14 68)(15 69)(16 158)(17 159)(18 160)(19 156)(20 157)(21 151)(22 152)(23 153)(24 154)(25 155)(26 56)(27 57)(28 58)(29 59)(30 60)(31 143)(32 144)(33 145)(34 141)(35 142)(41 55)(42 51)(43 52)(44 53)(45 54)(46 108)(47 109)(48 110)(49 106)(50 107)(61 97)(62 98)(63 99)(64 100)(65 96)(71 84)(72 85)(73 81)(74 82)(75 83)(76 93)(77 94)(78 95)(79 91)(80 92)(86 148)(87 149)(88 150)(89 146)(90 147)(101 137)(102 138)(103 139)(104 140)(105 136)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)
(1 89 12 134)(2 90 13 135)(3 86 14 131)(4 87 15 132)(5 88 11 133)(6 109 160 80)(7 110 156 76)(8 106 157 77)(9 107 158 78)(10 108 159 79)(16 95 130 50)(17 91 126 46)(18 92 127 47)(19 93 128 48)(20 94 129 49)(21 84 142 62)(22 85 143 63)(23 81 144 64)(24 82 145 65)(25 83 141 61)(26 122 42 105)(27 123 43 101)(28 124 44 102)(29 125 45 103)(30 121 41 104)(31 99 152 72)(32 100 153 73)(33 96 154 74)(34 97 155 75)(35 98 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(51 136 56 114)(52 137 57 115)(53 138 58 111)(54 139 59 112)(55 140 60 113)
(1 32 60 20)(2 33 56 16)(3 34 57 17)(4 35 58 18)(5 31 59 19)(6 69 21 44)(7 70 22 45)(8 66 23 41)(9 67 24 42)(10 68 25 43)(11 152 54 128)(12 153 55 129)(13 154 51 130)(14 155 52 126)(15 151 53 127)(26 158 38 145)(27 159 39 141)(28 160 40 142)(29 156 36 143)(30 157 37 144)(46 148 75 123)(47 149 71 124)(48 150 72 125)(49 146 73 121)(50 147 74 122)(61 137 79 131)(62 138 80 132)(63 139 76 133)(64 140 77 134)(65 136 78 135)(81 113 106 89)(82 114 107 90)(83 115 108 86)(84 111 109 87)(85 112 110 88)(91 119 97 101)(92 120 98 102)(93 116 99 103)(94 117 100 104)(95 118 96 105)
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)(6,142)(7,143)(8,144)(9,145)(10,141)(11,36)(12,37)(13,38)(14,39)(15,40)(16,154)(17,155)(18,151)(19,152)(20,153)(21,160)(22,156)(23,157)(24,158)(25,159)(26,51)(27,52)(28,53)(29,54)(30,55)(31,128)(32,129)(33,130)(34,126)(35,127)(41,60)(42,56)(43,57)(44,58)(45,59)(46,79)(47,80)(48,76)(49,77)(50,78)(61,75)(62,71)(63,72)(64,73)(65,74)(81,100)(82,96)(83,97)(84,98)(85,99)(86,137)(87,138)(88,139)(89,140)(90,136)(91,108)(92,109)(93,110)(94,106)(95,107)(101,148)(102,149)(103,150)(104,146)(105,147)(111,132)(112,133)(113,134)(114,135)(115,131)(116,125)(117,121)(118,122)(119,123)(120,124), (1,30)(2,26)(3,27)(4,28)(5,29)(6,151)(7,152)(8,153)(9,154)(10,155)(11,45)(12,41)(13,42)(14,43)(15,44)(16,145)(17,141)(18,142)(19,143)(20,144)(21,127)(22,128)(23,129)(24,130)(25,126)(31,156)(32,157)(33,158)(34,159)(35,160)(36,59)(37,60)(38,56)(39,57)(40,58)(46,83)(47,84)(48,85)(49,81)(50,82)(51,67)(52,68)(53,69)(54,70)(55,66)(61,91)(62,92)(63,93)(64,94)(65,95)(71,109)(72,110)(73,106)(74,107)(75,108)(76,99)(77,100)(78,96)(79,97)(80,98)(86,123)(87,124)(88,125)(89,121)(90,122)(101,131)(102,132)(103,133)(104,134)(105,135)(111,149)(112,150)(113,146)(114,147)(115,148)(116,139)(117,140)(118,136)(119,137)(120,138), (1,37)(2,38)(3,39)(4,40)(5,36)(6,127)(7,128)(8,129)(9,130)(10,126)(11,70)(12,66)(13,67)(14,68)(15,69)(16,158)(17,159)(18,160)(19,156)(20,157)(21,151)(22,152)(23,153)(24,154)(25,155)(26,56)(27,57)(28,58)(29,59)(30,60)(31,143)(32,144)(33,145)(34,141)(35,142)(41,55)(42,51)(43,52)(44,53)(45,54)(46,108)(47,109)(48,110)(49,106)(50,107)(61,97)(62,98)(63,99)(64,100)(65,96)(71,84)(72,85)(73,81)(74,82)(75,83)(76,93)(77,94)(78,95)(79,91)(80,92)(86,148)(87,149)(88,150)(89,146)(90,147)(101,137)(102,138)(103,139)(104,140)(105,136)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132), (1,89,12,134)(2,90,13,135)(3,86,14,131)(4,87,15,132)(5,88,11,133)(6,109,160,80)(7,110,156,76)(8,106,157,77)(9,107,158,78)(10,108,159,79)(16,95,130,50)(17,91,126,46)(18,92,127,47)(19,93,128,48)(20,94,129,49)(21,84,142,62)(22,85,143,63)(23,81,144,64)(24,82,145,65)(25,83,141,61)(26,122,42,105)(27,123,43,101)(28,124,44,102)(29,125,45,103)(30,121,41,104)(31,99,152,72)(32,100,153,73)(33,96,154,74)(34,97,155,75)(35,98,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(51,136,56,114)(52,137,57,115)(53,138,58,111)(54,139,59,112)(55,140,60,113), (1,32,60,20)(2,33,56,16)(3,34,57,17)(4,35,58,18)(5,31,59,19)(6,69,21,44)(7,70,22,45)(8,66,23,41)(9,67,24,42)(10,68,25,43)(11,152,54,128)(12,153,55,129)(13,154,51,130)(14,155,52,126)(15,151,53,127)(26,158,38,145)(27,159,39,141)(28,160,40,142)(29,156,36,143)(30,157,37,144)(46,148,75,123)(47,149,71,124)(48,150,72,125)(49,146,73,121)(50,147,74,122)(61,137,79,131)(62,138,80,132)(63,139,76,133)(64,140,77,134)(65,136,78,135)(81,113,106,89)(82,114,107,90)(83,115,108,86)(84,111,109,87)(85,112,110,88)(91,119,97,101)(92,120,98,102)(93,116,99,103)(94,117,100,104)(95,118,96,105)>;
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)(6,142)(7,143)(8,144)(9,145)(10,141)(11,36)(12,37)(13,38)(14,39)(15,40)(16,154)(17,155)(18,151)(19,152)(20,153)(21,160)(22,156)(23,157)(24,158)(25,159)(26,51)(27,52)(28,53)(29,54)(30,55)(31,128)(32,129)(33,130)(34,126)(35,127)(41,60)(42,56)(43,57)(44,58)(45,59)(46,79)(47,80)(48,76)(49,77)(50,78)(61,75)(62,71)(63,72)(64,73)(65,74)(81,100)(82,96)(83,97)(84,98)(85,99)(86,137)(87,138)(88,139)(89,140)(90,136)(91,108)(92,109)(93,110)(94,106)(95,107)(101,148)(102,149)(103,150)(104,146)(105,147)(111,132)(112,133)(113,134)(114,135)(115,131)(116,125)(117,121)(118,122)(119,123)(120,124), (1,30)(2,26)(3,27)(4,28)(5,29)(6,151)(7,152)(8,153)(9,154)(10,155)(11,45)(12,41)(13,42)(14,43)(15,44)(16,145)(17,141)(18,142)(19,143)(20,144)(21,127)(22,128)(23,129)(24,130)(25,126)(31,156)(32,157)(33,158)(34,159)(35,160)(36,59)(37,60)(38,56)(39,57)(40,58)(46,83)(47,84)(48,85)(49,81)(50,82)(51,67)(52,68)(53,69)(54,70)(55,66)(61,91)(62,92)(63,93)(64,94)(65,95)(71,109)(72,110)(73,106)(74,107)(75,108)(76,99)(77,100)(78,96)(79,97)(80,98)(86,123)(87,124)(88,125)(89,121)(90,122)(101,131)(102,132)(103,133)(104,134)(105,135)(111,149)(112,150)(113,146)(114,147)(115,148)(116,139)(117,140)(118,136)(119,137)(120,138), (1,37)(2,38)(3,39)(4,40)(5,36)(6,127)(7,128)(8,129)(9,130)(10,126)(11,70)(12,66)(13,67)(14,68)(15,69)(16,158)(17,159)(18,160)(19,156)(20,157)(21,151)(22,152)(23,153)(24,154)(25,155)(26,56)(27,57)(28,58)(29,59)(30,60)(31,143)(32,144)(33,145)(34,141)(35,142)(41,55)(42,51)(43,52)(44,53)(45,54)(46,108)(47,109)(48,110)(49,106)(50,107)(61,97)(62,98)(63,99)(64,100)(65,96)(71,84)(72,85)(73,81)(74,82)(75,83)(76,93)(77,94)(78,95)(79,91)(80,92)(86,148)(87,149)(88,150)(89,146)(90,147)(101,137)(102,138)(103,139)(104,140)(105,136)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132), (1,89,12,134)(2,90,13,135)(3,86,14,131)(4,87,15,132)(5,88,11,133)(6,109,160,80)(7,110,156,76)(8,106,157,77)(9,107,158,78)(10,108,159,79)(16,95,130,50)(17,91,126,46)(18,92,127,47)(19,93,128,48)(20,94,129,49)(21,84,142,62)(22,85,143,63)(23,81,144,64)(24,82,145,65)(25,83,141,61)(26,122,42,105)(27,123,43,101)(28,124,44,102)(29,125,45,103)(30,121,41,104)(31,99,152,72)(32,100,153,73)(33,96,154,74)(34,97,155,75)(35,98,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(51,136,56,114)(52,137,57,115)(53,138,58,111)(54,139,59,112)(55,140,60,113), (1,32,60,20)(2,33,56,16)(3,34,57,17)(4,35,58,18)(5,31,59,19)(6,69,21,44)(7,70,22,45)(8,66,23,41)(9,67,24,42)(10,68,25,43)(11,152,54,128)(12,153,55,129)(13,154,51,130)(14,155,52,126)(15,151,53,127)(26,158,38,145)(27,159,39,141)(28,160,40,142)(29,156,36,143)(30,157,37,144)(46,148,75,123)(47,149,71,124)(48,150,72,125)(49,146,73,121)(50,147,74,122)(61,137,79,131)(62,138,80,132)(63,139,76,133)(64,140,77,134)(65,136,78,135)(81,113,106,89)(82,114,107,90)(83,115,108,86)(84,111,109,87)(85,112,110,88)(91,119,97,101)(92,120,98,102)(93,116,99,103)(94,117,100,104)(95,118,96,105) );
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),(6,142),(7,143),(8,144),(9,145),(10,141),(11,36),(12,37),(13,38),(14,39),(15,40),(16,154),(17,155),(18,151),(19,152),(20,153),(21,160),(22,156),(23,157),(24,158),(25,159),(26,51),(27,52),(28,53),(29,54),(30,55),(31,128),(32,129),(33,130),(34,126),(35,127),(41,60),(42,56),(43,57),(44,58),(45,59),(46,79),(47,80),(48,76),(49,77),(50,78),(61,75),(62,71),(63,72),(64,73),(65,74),(81,100),(82,96),(83,97),(84,98),(85,99),(86,137),(87,138),(88,139),(89,140),(90,136),(91,108),(92,109),(93,110),(94,106),(95,107),(101,148),(102,149),(103,150),(104,146),(105,147),(111,132),(112,133),(113,134),(114,135),(115,131),(116,125),(117,121),(118,122),(119,123),(120,124)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,151),(7,152),(8,153),(9,154),(10,155),(11,45),(12,41),(13,42),(14,43),(15,44),(16,145),(17,141),(18,142),(19,143),(20,144),(21,127),(22,128),(23,129),(24,130),(25,126),(31,156),(32,157),(33,158),(34,159),(35,160),(36,59),(37,60),(38,56),(39,57),(40,58),(46,83),(47,84),(48,85),(49,81),(50,82),(51,67),(52,68),(53,69),(54,70),(55,66),(61,91),(62,92),(63,93),(64,94),(65,95),(71,109),(72,110),(73,106),(74,107),(75,108),(76,99),(77,100),(78,96),(79,97),(80,98),(86,123),(87,124),(88,125),(89,121),(90,122),(101,131),(102,132),(103,133),(104,134),(105,135),(111,149),(112,150),(113,146),(114,147),(115,148),(116,139),(117,140),(118,136),(119,137),(120,138)], [(1,37),(2,38),(3,39),(4,40),(5,36),(6,127),(7,128),(8,129),(9,130),(10,126),(11,70),(12,66),(13,67),(14,68),(15,69),(16,158),(17,159),(18,160),(19,156),(20,157),(21,151),(22,152),(23,153),(24,154),(25,155),(26,56),(27,57),(28,58),(29,59),(30,60),(31,143),(32,144),(33,145),(34,141),(35,142),(41,55),(42,51),(43,52),(44,53),(45,54),(46,108),(47,109),(48,110),(49,106),(50,107),(61,97),(62,98),(63,99),(64,100),(65,96),(71,84),(72,85),(73,81),(74,82),(75,83),(76,93),(77,94),(78,95),(79,91),(80,92),(86,148),(87,149),(88,150),(89,146),(90,147),(101,137),(102,138),(103,139),(104,140),(105,136),(111,124),(112,125),(113,121),(114,122),(115,123),(116,133),(117,134),(118,135),(119,131),(120,132)], [(1,89,12,134),(2,90,13,135),(3,86,14,131),(4,87,15,132),(5,88,11,133),(6,109,160,80),(7,110,156,76),(8,106,157,77),(9,107,158,78),(10,108,159,79),(16,95,130,50),(17,91,126,46),(18,92,127,47),(19,93,128,48),(20,94,129,49),(21,84,142,62),(22,85,143,63),(23,81,144,64),(24,82,145,65),(25,83,141,61),(26,122,42,105),(27,123,43,101),(28,124,44,102),(29,125,45,103),(30,121,41,104),(31,99,152,72),(32,100,153,73),(33,96,154,74),(34,97,155,75),(35,98,151,71),(36,150,70,116),(37,146,66,117),(38,147,67,118),(39,148,68,119),(40,149,69,120),(51,136,56,114),(52,137,57,115),(53,138,58,111),(54,139,59,112),(55,140,60,113)], [(1,32,60,20),(2,33,56,16),(3,34,57,17),(4,35,58,18),(5,31,59,19),(6,69,21,44),(7,70,22,45),(8,66,23,41),(9,67,24,42),(10,68,25,43),(11,152,54,128),(12,153,55,129),(13,154,51,130),(14,155,52,126),(15,151,53,127),(26,158,38,145),(27,159,39,141),(28,160,40,142),(29,156,36,143),(30,157,37,144),(46,148,75,123),(47,149,71,124),(48,150,72,125),(49,146,73,121),(50,147,74,122),(61,137,79,131),(62,138,80,132),(63,139,76,133),(64,140,77,134),(65,136,78,135),(81,113,106,89),(82,114,107,90),(83,115,108,86),(84,111,109,87),(85,112,110,88),(91,119,97,101),(92,120,98,102),(93,116,99,103),(94,117,100,104),(95,118,96,105)]])
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