direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.46D4, C4.Q8⋊9C10, C22⋊C8⋊10C10, C4⋊D4.6C10, (C2×C20).337D4, D4⋊C4⋊12C10, C23.46(C5×D4), C2.12(C10×SD16), C10.92(C2×SD16), (C2×C10).36SD16, C20.318(C4○D4), C22.8(C5×SD16), (C2×C20).937C23, (C2×C40).305C22, (C22×C10).168D4, C22.102(D4×C10), C10.142(C8⋊C22), (D4×C10).196C22, (C22×C20).429C22, C10.96(C22.D4), (C10×C4⋊C4)⋊39C2, (C2×C4⋊C4)⋊12C10, (C5×C4.Q8)⋊24C2, C4.30(C5×C4○D4), (C2×C4).38(C5×D4), C4⋊C4.58(C2×C10), (C5×C22⋊C8)⋊27C2, (C2×C8).42(C2×C10), C2.17(C5×C8⋊C22), (C5×D4⋊C4)⋊36C2, (C2×D4).19(C2×C10), (C5×C4⋊D4).16C2, (C2×C10).658(C2×D4), (C5×C4⋊C4).381C22, (C22×C4).47(C2×C10), (C2×C4).112(C22×C10), C2.12(C5×C22.D4), SmallGroup(320,982)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.46D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >
Subgroups: 226 in 114 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C23.46D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, C5×C22⋊C8, C5×D4⋊C4, C5×C4.Q8, C10×C4⋊C4, C5×C4⋊D4, C5×C23.46D4
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C4○D4, C2×C10, C22.D4, C2×SD16, C8⋊C22, C5×D4, C22×C10, C23.46D4, C5×SD16, D4×C10, C5×C4○D4, C5×C22.D4, C10×SD16, C5×C8⋊C22, C5×C23.46D4
►On 160 points
Generators in S160
(1 62 20 54 12)(2 63 21 55 13)(3 64 22 56 14)(4 57 23 49 15)(5 58 24 50 16)(6 59 17 51 9)(7 60 18 52 10)(8 61 19 53 11)(25 85 75 33 67)(26 86 76 34 68)(27 87 77 35 69)(28 88 78 36 70)(29 81 79 37 71)(30 82 80 38 72)(31 83 73 39 65)(32 84 74 40 66)(41 155 117 147 109)(42 156 118 148 110)(43 157 119 149 111)(44 158 120 150 112)(45 159 113 151 105)(46 160 114 152 106)(47 153 115 145 107)(48 154 116 146 108)(89 127 138 100 130)(90 128 139 101 131)(91 121 140 102 132)(92 122 141 103 133)(93 123 142 104 134)(94 124 143 97 135)(95 125 144 98 136)(96 126 137 99 129)
(1 111)(2 132)(3 105)(4 134)(5 107)(6 136)(7 109)(8 130)(9 98)(10 147)(11 100)(12 149)(13 102)(14 151)(15 104)(16 145)(17 125)(18 155)(19 127)(20 157)(21 121)(22 159)(23 123)(24 153)(25 129)(26 110)(27 131)(28 112)(29 133)(30 106)(31 135)(32 108)(33 137)(34 118)(35 139)(36 120)(37 141)(38 114)(39 143)(40 116)(41 60)(42 86)(43 62)(44 88)(45 64)(46 82)(47 58)(48 84)(49 142)(50 115)(51 144)(52 117)(53 138)(54 119)(55 140)(56 113)(57 93)(59 95)(61 89)(63 91)(65 97)(66 146)(67 99)(68 148)(69 101)(70 150)(71 103)(72 152)(73 124)(74 154)(75 126)(76 156)(77 128)(78 158)(79 122)(80 160)(81 92)(83 94)(85 96)(87 90)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 81)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 133)(106 134)(107 135)(108 136)(109 129)(110 130)(111 131)(112 132)(113 141)(114 142)(115 143)(116 144)(117 137)(118 138)(119 139)(120 140)(121 158)(122 159)(123 160)(124 153)(125 154)(126 155)(127 156)(128 157)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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)
(2 30)(3 7)(4 28)(6 26)(8 32)(9 68)(10 14)(11 66)(13 72)(15 70)(17 76)(18 22)(19 74)(21 80)(23 78)(25 29)(33 37)(34 51)(36 49)(38 55)(40 53)(41 92)(42 48)(43 90)(44 46)(45 96)(47 94)(52 56)(57 88)(59 86)(60 64)(61 84)(63 82)(67 71)(75 79)(81 85)(89 95)(91 93)(97 145)(98 100)(99 151)(101 149)(102 104)(103 147)(105 129)(106 112)(107 135)(108 110)(109 133)(111 131)(113 137)(114 120)(115 143)(116 118)(117 141)(119 139)(121 123)(122 155)(124 153)(125 127)(126 159)(128 157)(130 136)(132 134)(138 144)(140 142)(146 148)(150 152)(154 156)(158 160)
G:=sub<Sym(160)| (1,62,20,54,12)(2,63,21,55,13)(3,64,22,56,14)(4,57,23,49,15)(5,58,24,50,16)(6,59,17,51,9)(7,60,18,52,10)(8,61,19,53,11)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,155,117,147,109)(42,156,118,148,110)(43,157,119,149,111)(44,158,120,150,112)(45,159,113,151,105)(46,160,114,152,106)(47,153,115,145,107)(48,154,116,146,108)(89,127,138,100,130)(90,128,139,101,131)(91,121,140,102,132)(92,122,141,103,133)(93,123,142,104,134)(94,124,143,97,135)(95,125,144,98,136)(96,126,137,99,129), (1,111)(2,132)(3,105)(4,134)(5,107)(6,136)(7,109)(8,130)(9,98)(10,147)(11,100)(12,149)(13,102)(14,151)(15,104)(16,145)(17,125)(18,155)(19,127)(20,157)(21,121)(22,159)(23,123)(24,153)(25,129)(26,110)(27,131)(28,112)(29,133)(30,106)(31,135)(32,108)(33,137)(34,118)(35,139)(36,120)(37,141)(38,114)(39,143)(40,116)(41,60)(42,86)(43,62)(44,88)(45,64)(46,82)(47,58)(48,84)(49,142)(50,115)(51,144)(52,117)(53,138)(54,119)(55,140)(56,113)(57,93)(59,95)(61,89)(63,91)(65,97)(66,146)(67,99)(68,148)(69,101)(70,150)(71,103)(72,152)(73,124)(74,154)(75,126)(76,156)(77,128)(78,158)(79,122)(80,160)(81,92)(83,94)(85,96)(87,90), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,133)(106,134)(107,135)(108,136)(109,129)(110,130)(111,131)(112,132)(113,141)(114,142)(115,143)(116,144)(117,137)(118,138)(119,139)(120,140)(121,158)(122,159)(123,160)(124,153)(125,154)(126,155)(127,156)(128,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (2,30)(3,7)(4,28)(6,26)(8,32)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,92)(42,48)(43,90)(44,46)(45,96)(47,94)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,95)(91,93)(97,145)(98,100)(99,151)(101,149)(102,104)(103,147)(105,129)(106,112)(107,135)(108,110)(109,133)(111,131)(113,137)(114,120)(115,143)(116,118)(117,141)(119,139)(121,123)(122,155)(124,153)(125,127)(126,159)(128,157)(130,136)(132,134)(138,144)(140,142)(146,148)(150,152)(154,156)(158,160)>;
G:=Group( (1,62,20,54,12)(2,63,21,55,13)(3,64,22,56,14)(4,57,23,49,15)(5,58,24,50,16)(6,59,17,51,9)(7,60,18,52,10)(8,61,19,53,11)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,155,117,147,109)(42,156,118,148,110)(43,157,119,149,111)(44,158,120,150,112)(45,159,113,151,105)(46,160,114,152,106)(47,153,115,145,107)(48,154,116,146,108)(89,127,138,100,130)(90,128,139,101,131)(91,121,140,102,132)(92,122,141,103,133)(93,123,142,104,134)(94,124,143,97,135)(95,125,144,98,136)(96,126,137,99,129), (1,111)(2,132)(3,105)(4,134)(5,107)(6,136)(7,109)(8,130)(9,98)(10,147)(11,100)(12,149)(13,102)(14,151)(15,104)(16,145)(17,125)(18,155)(19,127)(20,157)(21,121)(22,159)(23,123)(24,153)(25,129)(26,110)(27,131)(28,112)(29,133)(30,106)(31,135)(32,108)(33,137)(34,118)(35,139)(36,120)(37,141)(38,114)(39,143)(40,116)(41,60)(42,86)(43,62)(44,88)(45,64)(46,82)(47,58)(48,84)(49,142)(50,115)(51,144)(52,117)(53,138)(54,119)(55,140)(56,113)(57,93)(59,95)(61,89)(63,91)(65,97)(66,146)(67,99)(68,148)(69,101)(70,150)(71,103)(72,152)(73,124)(74,154)(75,126)(76,156)(77,128)(78,158)(79,122)(80,160)(81,92)(83,94)(85,96)(87,90), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,133)(106,134)(107,135)(108,136)(109,129)(110,130)(111,131)(112,132)(113,141)(114,142)(115,143)(116,144)(117,137)(118,138)(119,139)(120,140)(121,158)(122,159)(123,160)(124,153)(125,154)(126,155)(127,156)(128,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (2,30)(3,7)(4,28)(6,26)(8,32)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,92)(42,48)(43,90)(44,46)(45,96)(47,94)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,95)(91,93)(97,145)(98,100)(99,151)(101,149)(102,104)(103,147)(105,129)(106,112)(107,135)(108,110)(109,133)(111,131)(113,137)(114,120)(115,143)(116,118)(117,141)(119,139)(121,123)(122,155)(124,153)(125,127)(126,159)(128,157)(130,136)(132,134)(138,144)(140,142)(146,148)(150,152)(154,156)(158,160) );
G=PermutationGroup([[(1,62,20,54,12),(2,63,21,55,13),(3,64,22,56,14),(4,57,23,49,15),(5,58,24,50,16),(6,59,17,51,9),(7,60,18,52,10),(8,61,19,53,11),(25,85,75,33,67),(26,86,76,34,68),(27,87,77,35,69),(28,88,78,36,70),(29,81,79,37,71),(30,82,80,38,72),(31,83,73,39,65),(32,84,74,40,66),(41,155,117,147,109),(42,156,118,148,110),(43,157,119,149,111),(44,158,120,150,112),(45,159,113,151,105),(46,160,114,152,106),(47,153,115,145,107),(48,154,116,146,108),(89,127,138,100,130),(90,128,139,101,131),(91,121,140,102,132),(92,122,141,103,133),(93,123,142,104,134),(94,124,143,97,135),(95,125,144,98,136),(96,126,137,99,129)], [(1,111),(2,132),(3,105),(4,134),(5,107),(6,136),(7,109),(8,130),(9,98),(10,147),(11,100),(12,149),(13,102),(14,151),(15,104),(16,145),(17,125),(18,155),(19,127),(20,157),(21,121),(22,159),(23,123),(24,153),(25,129),(26,110),(27,131),(28,112),(29,133),(30,106),(31,135),(32,108),(33,137),(34,118),(35,139),(36,120),(37,141),(38,114),(39,143),(40,116),(41,60),(42,86),(43,62),(44,88),(45,64),(46,82),(47,58),(48,84),(49,142),(50,115),(51,144),(52,117),(53,138),(54,119),(55,140),(56,113),(57,93),(59,95),(61,89),(63,91),(65,97),(66,146),(67,99),(68,148),(69,101),(70,150),(71,103),(72,152),(73,124),(74,154),(75,126),(76,156),(77,128),(78,158),(79,122),(80,160),(81,92),(83,94),(85,96),(87,90)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51),(41,96),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,81),(97,145),(98,146),(99,147),(100,148),(101,149),(102,150),(103,151),(104,152),(105,133),(106,134),(107,135),(108,136),(109,129),(110,130),(111,131),(112,132),(113,141),(114,142),(115,143),(116,144),(117,137),(118,138),(119,139),(120,140),(121,158),(122,159),(123,160),(124,153),(125,154),(126,155),(127,156),(128,157)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(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)], [(2,30),(3,7),(4,28),(6,26),(8,32),(9,68),(10,14),(11,66),(13,72),(15,70),(17,76),(18,22),(19,74),(21,80),(23,78),(25,29),(33,37),(34,51),(36,49),(38,55),(40,53),(41,92),(42,48),(43,90),(44,46),(45,96),(47,94),(52,56),(57,88),(59,86),(60,64),(61,84),(63,82),(67,71),(75,79),(81,85),(89,95),(91,93),(97,145),(98,100),(99,151),(101,149),(102,104),(103,147),(105,129),(106,112),(107,135),(108,110),(109,133),(111,131),(113,137),(114,120),(115,143),(116,118),(117,141),(119,139),(121,123),(122,155),(124,153),(125,127),(126,159),(128,157),(130,136),(132,134),(138,144),(140,142),(146,148),(150,152),(154,156),(158,160)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | SD16 | C5×D4 | C5×D4 | C5×C4○D4 | C5×SD16 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C23.46D4 | C5×C22⋊C8 | C5×D4⋊C4 | C5×C4.Q8 | C10×C4⋊C4 | C5×C4⋊D4 | C23.46D4 | C22⋊C8 | D4⋊C4 | C4.Q8 | C2×C4⋊C4 | C4⋊D4 | C2×C20 | C22×C10 | C20 | C2×C10 | C2×C4 | C23 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C23.46D4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 15 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,9,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C23.46D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{46}D_4 % in TeX
G:=Group("C5xC2^3.46D4"); // GroupNames label
G:=SmallGroup(320,982);
// by ID
G=gap.SmallGroup(320,982);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1066,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=f^2=1,e^4=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=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*e^3>;
// generators/relations