direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.23D10, C24.70D10, (C23×C4)⋊4D5, (C23×C20)⋊4C2, (C22×C4)⋊43D10, (C2×C10).287C24, (C2×C20).704C23, (C22×C20)⋊56C22, C10.133(C22×D4), (C22×C10).205D4, C23.91(C5⋊D4), C23.D5⋊55C22, D10⋊C4⋊41C22, C22.82(C4○D20), C10.D4⋊44C22, C10⋊4(C22.D4), (C23×D5).74C22, C23.233(C22×D5), C22.302(C23×D5), (C22×C10).416C23, (C23×C10).109C22, (C2×Dic5).149C23, (C22×D5).125C23, (C22×Dic5).161C22, C2.70(C2×C4○D20), C10.62(C2×C4○D4), C2.6(C22×C5⋊D4), C5⋊5(C2×C22.D4), (C2×C10).574(C2×D4), (C2×C23.D5)⋊22C2, (C2×D10⋊C4)⋊13C2, (C2×C10.D4)⋊18C2, (C2×C4).657(C22×D5), (C22×C5⋊D4).13C2, C22.103(C2×C5⋊D4), (C2×C10).113(C4○D4), (C2×C5⋊D4).144C22, SmallGroup(320,1461)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.23D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 1118 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C22.D4, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C23×D5, C23×C10, C2×C10.D4, C2×D10⋊C4, C23.23D10, C2×C23.D5, C22×C5⋊D4, C23×C20, C2×C23.23D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22.D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C22.D4, C4○D20, C2×C5⋊D4, C23×D5, C23.23D10, C2×C4○D20, C22×C5⋊D4, C2×C23.23D10
►On 160 points
Generators in S160
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 160)(22 141)(23 142)(24 143)(25 144)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 101)(61 157)(62 158)(63 159)(64 160)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 121)
(1 137)(2 138)(3 139)(4 140)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 160)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 119)(62 120)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(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 10 127 136)(2 135 128 9)(3 8 129 134)(4 133 130 7)(5 6 131 132)(11 20 137 126)(12 125 138 19)(13 18 139 124)(14 123 140 17)(15 16 121 122)(21 32 100 91)(22 90 81 31)(23 30 82 89)(24 88 83 29)(25 28 84 87)(26 86 85 27)(33 40 92 99)(34 98 93 39)(35 38 94 97)(36 96 95 37)(41 52 150 141)(42 160 151 51)(43 50 152 159)(44 158 153 49)(45 48 154 157)(46 156 155 47)(53 60 142 149)(54 148 143 59)(55 58 144 147)(56 146 145 57)(61 116 109 68)(62 67 110 115)(63 114 111 66)(64 65 112 113)(69 108 117 80)(70 79 118 107)(71 106 119 78)(72 77 120 105)(73 104 101 76)(74 75 102 103)
G:=sub<Sym(160)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,160)(22,141)(23,142)(24,143)(25,144)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,101)(61,157)(62,158)(63,159)(64,160)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,121), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,160)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (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), (1,10,127,136)(2,135,128,9)(3,8,129,134)(4,133,130,7)(5,6,131,132)(11,20,137,126)(12,125,138,19)(13,18,139,124)(14,123,140,17)(15,16,121,122)(21,32,100,91)(22,90,81,31)(23,30,82,89)(24,88,83,29)(25,28,84,87)(26,86,85,27)(33,40,92,99)(34,98,93,39)(35,38,94,97)(36,96,95,37)(41,52,150,141)(42,160,151,51)(43,50,152,159)(44,158,153,49)(45,48,154,157)(46,156,155,47)(53,60,142,149)(54,148,143,59)(55,58,144,147)(56,146,145,57)(61,116,109,68)(62,67,110,115)(63,114,111,66)(64,65,112,113)(69,108,117,80)(70,79,118,107)(71,106,119,78)(72,77,120,105)(73,104,101,76)(74,75,102,103)>;
G:=Group( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,160)(22,141)(23,142)(24,143)(25,144)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,101)(61,157)(62,158)(63,159)(64,160)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,121), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,160)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (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), (1,10,127,136)(2,135,128,9)(3,8,129,134)(4,133,130,7)(5,6,131,132)(11,20,137,126)(12,125,138,19)(13,18,139,124)(14,123,140,17)(15,16,121,122)(21,32,100,91)(22,90,81,31)(23,30,82,89)(24,88,83,29)(25,28,84,87)(26,86,85,27)(33,40,92,99)(34,98,93,39)(35,38,94,97)(36,96,95,37)(41,52,150,141)(42,160,151,51)(43,50,152,159)(44,158,153,49)(45,48,154,157)(46,156,155,47)(53,60,142,149)(54,148,143,59)(55,58,144,147)(56,146,145,57)(61,116,109,68)(62,67,110,115)(63,114,111,66)(64,65,112,113)(69,108,117,80)(70,79,118,107)(71,106,119,78)(72,77,120,105)(73,104,101,76)(74,75,102,103) );
G=PermutationGroup([[(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,160),(22,141),(23,142),(24,143),(25,144),(26,145),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,21),(16,22),(17,23),(18,24),(19,25),(20,26),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,101),(61,157),(62,158),(63,159),(64,160),(65,141),(66,142),(67,143),(68,144),(69,145),(70,146),(71,147),(72,148),(73,149),(74,150),(75,151),(76,152),(77,153),(78,154),(79,155),(80,156),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,129),(89,130),(90,131),(91,132),(92,133),(93,134),(94,135),(95,136),(96,137),(97,138),(98,139),(99,140),(100,121)], [(1,137),(2,138),(3,139),(4,140),(5,121),(6,122),(7,123),(8,124),(9,125),(10,126),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,160),(42,141),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,155),(57,156),(58,157),(59,158),(60,159),(61,119),(62,120),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(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)], [(1,10,127,136),(2,135,128,9),(3,8,129,134),(4,133,130,7),(5,6,131,132),(11,20,137,126),(12,125,138,19),(13,18,139,124),(14,123,140,17),(15,16,121,122),(21,32,100,91),(22,90,81,31),(23,30,82,89),(24,88,83,29),(25,28,84,87),(26,86,85,27),(33,40,92,99),(34,98,93,39),(35,38,94,97),(36,96,95,37),(41,52,150,141),(42,160,151,51),(43,50,152,159),(44,158,153,49),(45,48,154,157),(46,156,155,47),(53,60,142,149),(54,148,143,59),(55,58,144,147),(56,146,145,57),(61,116,109,68),(62,67,110,115),(63,114,111,66),(64,65,112,113),(69,108,117,80),(70,79,118,107),(71,106,119,78),(72,77,120,105),(73,104,101,76),(74,75,102,103)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4○D20 |
| kernel | C2×C23.23D10 | C2×C10.D4 | C2×D10⋊C4 | C23.23D10 | C2×C23.D5 | C22×C5⋊D4 | C23×C20 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 4 | 2 | 8 | 12 | 2 | 16 | 32 |
Matrix representation of C2×C23.23D10 ►in GL5(𝔽41)
| 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 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 16 | 5 |
| 0 | 0 | 0 | 31 | 25 |
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 38 | 0 | 0 |
| 0 | 3 | 38 | 0 | 0 |
| 0 | 0 | 0 | 20 | 37 |
| 0 | 0 | 0 | 8 | 21 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 24 | 3 | 0 | 0 |
| 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 20 | 27 |
| 0 | 0 | 0 | 8 | 21 |
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],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,16,31,0,0,0,5,25],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,17,3,0,0,0,38,38,0,0,0,0,0,20,8,0,0,0,37,21],[40,0,0,0,0,0,24,40,0,0,0,3,17,0,0,0,0,0,20,8,0,0,0,27,21] >;
C2×C23.23D10 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{23}D_{10} % in TeX
G:=Group("C2xC2^3.23D10"); // GroupNames label
G:=SmallGroup(320,1461);
// by ID
G=gap.SmallGroup(320,1461);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,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,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;
// generators/relations