direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.11D10, C24.50D10, C23.57(C4×D5), (C2×C10).24C24, C10.27(C23×C4), C10⋊3(C42⋊C2), (C2×C20).569C23, C22⋊C4.122D10, (C22×Dic5)⋊14C4, (C4×Dic5)⋊71C22, (C22×C4).311D10, (C23×Dic5).7C2, C22.16(C23×D5), C10.D4⋊56C22, (C23×C10).50C22, Dic5.52(C22×C4), C23.143(C22×D5), C23.D5.82C22, C22.63(D4⋊2D5), (C22×C20).349C22, (C22×C10).386C23, (C2×Dic5).369C23, (C22×Dic5).288C22, C2.8(D5×C22×C4), C5⋊3(C2×C42⋊C2), (C2×C4×Dic5)⋊28C2, C22.23(C2×C4×D5), C10.65(C2×C4○D4), C2.1(C2×D4⋊2D5), (C2×Dic5)⋊30(C2×C4), (C2×C22⋊C4).20D5, (C2×C10.D4)⋊33C2, (C10×C22⋊C4).25C2, (C2×C4).254(C22×D5), (C2×C23.D5).19C2, (C2×C10).165(C4○D4), (C2×C10).118(C22×C4), (C22×C10).144(C2×C4), (C5×C22⋊C4).132C22, SmallGroup(320,1152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.11D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 830 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C42⋊C2, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C23.11D10, C2×C4×Dic5, C2×C10.D4, C2×C23.D5, C10×C22⋊C4, C23×Dic5, C2×C23.11D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, C24, D10, C42⋊C2, C23×C4, C2×C4○D4, C4×D5, C22×D5, C2×C42⋊C2, C2×C4×D5, D4⋊2D5, C23×D5, C23.11D10, D5×C22×C4, C2×D4⋊2D5, C2×C23.11D10
►On 160 points
Generators in S160
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)
(1 43)(2 116)(3 45)(4 118)(5 47)(6 120)(7 49)(8 102)(9 51)(10 104)(11 53)(12 106)(13 55)(14 108)(15 57)(16 110)(17 59)(18 112)(19 41)(20 114)(21 125)(22 145)(23 127)(24 147)(25 129)(26 149)(27 131)(28 151)(29 133)(30 153)(31 135)(32 155)(33 137)(34 157)(35 139)(36 159)(37 121)(38 141)(39 123)(40 143)(42 84)(44 86)(46 88)(48 90)(50 92)(52 94)(54 96)(56 98)(58 100)(60 82)(61 140)(62 160)(63 122)(64 142)(65 124)(66 144)(67 126)(68 146)(69 128)(70 148)(71 130)(72 150)(73 132)(74 152)(75 134)(76 154)(77 136)(78 156)(79 138)(80 158)(81 111)(83 113)(85 115)(87 117)(89 119)(91 101)(93 103)(95 105)(97 107)(99 109)
(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 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 81)(18 82)(19 83)(20 84)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 61)(37 62)(38 63)(39 64)(40 65)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(121 160)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)
(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 38 95 73)(2 27 96 62)(3 36 97 71)(4 25 98 80)(5 34 99 69)(6 23 100 78)(7 32 81 67)(8 21 82 76)(9 30 83 65)(10 39 84 74)(11 28 85 63)(12 37 86 72)(13 26 87 61)(14 35 88 70)(15 24 89 79)(16 33 90 68)(17 22 91 77)(18 31 92 66)(19 40 93 75)(20 29 94 64)(41 124 103 153)(42 133 104 142)(43 122 105 151)(44 131 106 160)(45 140 107 149)(46 129 108 158)(47 138 109 147)(48 127 110 156)(49 136 111 145)(50 125 112 154)(51 134 113 143)(52 123 114 152)(53 132 115 141)(54 121 116 150)(55 130 117 159)(56 139 118 148)(57 128 119 157)(58 137 120 146)(59 126 101 155)(60 135 102 144)
G:=sub<Sym(160)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120), (1,43)(2,116)(3,45)(4,118)(5,47)(6,120)(7,49)(8,102)(9,51)(10,104)(11,53)(12,106)(13,55)(14,108)(15,57)(16,110)(17,59)(18,112)(19,41)(20,114)(21,125)(22,145)(23,127)(24,147)(25,129)(26,149)(27,131)(28,151)(29,133)(30,153)(31,135)(32,155)(33,137)(34,157)(35,139)(36,159)(37,121)(38,141)(39,123)(40,143)(42,84)(44,86)(46,88)(48,90)(50,92)(52,94)(54,96)(56,98)(58,100)(60,82)(61,140)(62,160)(63,122)(64,142)(65,124)(66,144)(67,126)(68,146)(69,128)(70,148)(71,130)(72,150)(73,132)(74,152)(75,134)(76,154)(77,136)(78,156)(79,138)(80,158)(81,111)(83,113)(85,115)(87,117)(89,119)(91,101)(93,103)(95,105)(97,107)(99,109), (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), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,81)(18,82)(19,83)(20,84)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(121,160)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159), (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), (1,38,95,73)(2,27,96,62)(3,36,97,71)(4,25,98,80)(5,34,99,69)(6,23,100,78)(7,32,81,67)(8,21,82,76)(9,30,83,65)(10,39,84,74)(11,28,85,63)(12,37,86,72)(13,26,87,61)(14,35,88,70)(15,24,89,79)(16,33,90,68)(17,22,91,77)(18,31,92,66)(19,40,93,75)(20,29,94,64)(41,124,103,153)(42,133,104,142)(43,122,105,151)(44,131,106,160)(45,140,107,149)(46,129,108,158)(47,138,109,147)(48,127,110,156)(49,136,111,145)(50,125,112,154)(51,134,113,143)(52,123,114,152)(53,132,115,141)(54,121,116,150)(55,130,117,159)(56,139,118,148)(57,128,119,157)(58,137,120,146)(59,126,101,155)(60,135,102,144)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120), (1,43)(2,116)(3,45)(4,118)(5,47)(6,120)(7,49)(8,102)(9,51)(10,104)(11,53)(12,106)(13,55)(14,108)(15,57)(16,110)(17,59)(18,112)(19,41)(20,114)(21,125)(22,145)(23,127)(24,147)(25,129)(26,149)(27,131)(28,151)(29,133)(30,153)(31,135)(32,155)(33,137)(34,157)(35,139)(36,159)(37,121)(38,141)(39,123)(40,143)(42,84)(44,86)(46,88)(48,90)(50,92)(52,94)(54,96)(56,98)(58,100)(60,82)(61,140)(62,160)(63,122)(64,142)(65,124)(66,144)(67,126)(68,146)(69,128)(70,148)(71,130)(72,150)(73,132)(74,152)(75,134)(76,154)(77,136)(78,156)(79,138)(80,158)(81,111)(83,113)(85,115)(87,117)(89,119)(91,101)(93,103)(95,105)(97,107)(99,109), (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), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,81)(18,82)(19,83)(20,84)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(121,160)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159), (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), (1,38,95,73)(2,27,96,62)(3,36,97,71)(4,25,98,80)(5,34,99,69)(6,23,100,78)(7,32,81,67)(8,21,82,76)(9,30,83,65)(10,39,84,74)(11,28,85,63)(12,37,86,72)(13,26,87,61)(14,35,88,70)(15,24,89,79)(16,33,90,68)(17,22,91,77)(18,31,92,66)(19,40,93,75)(20,29,94,64)(41,124,103,153)(42,133,104,142)(43,122,105,151)(44,131,106,160)(45,140,107,149)(46,129,108,158)(47,138,109,147)(48,127,110,156)(49,136,111,145)(50,125,112,154)(51,134,113,143)(52,123,114,152)(53,132,115,141)(54,121,116,150)(55,130,117,159)(56,139,118,148)(57,128,119,157)(58,137,120,146)(59,126,101,155)(60,135,102,144) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,141),(74,142),(75,143),(76,144),(77,145),(78,146),(79,147),(80,148),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120)], [(1,43),(2,116),(3,45),(4,118),(5,47),(6,120),(7,49),(8,102),(9,51),(10,104),(11,53),(12,106),(13,55),(14,108),(15,57),(16,110),(17,59),(18,112),(19,41),(20,114),(21,125),(22,145),(23,127),(24,147),(25,129),(26,149),(27,131),(28,151),(29,133),(30,153),(31,135),(32,155),(33,137),(34,157),(35,139),(36,159),(37,121),(38,141),(39,123),(40,143),(42,84),(44,86),(46,88),(48,90),(50,92),(52,94),(54,96),(56,98),(58,100),(60,82),(61,140),(62,160),(63,122),(64,142),(65,124),(66,144),(67,126),(68,146),(69,128),(70,148),(71,130),(72,150),(73,132),(74,152),(75,134),(76,154),(77,136),(78,156),(79,138),(80,158),(81,111),(83,113),(85,115),(87,117),(89,119),(91,101),(93,103),(95,105),(97,107),(99,109)], [(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)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,81),(18,82),(19,83),(20,84),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,61),(37,62),(38,63),(39,64),(40,65),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(121,160),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152),(134,153),(135,154),(136,155),(137,156),(138,157),(139,158),(140,159)], [(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)], [(1,38,95,73),(2,27,96,62),(3,36,97,71),(4,25,98,80),(5,34,99,69),(6,23,100,78),(7,32,81,67),(8,21,82,76),(9,30,83,65),(10,39,84,74),(11,28,85,63),(12,37,86,72),(13,26,87,61),(14,35,88,70),(15,24,89,79),(16,33,90,68),(17,22,91,77),(18,31,92,66),(19,40,93,75),(20,29,94,64),(41,124,103,153),(42,133,104,142),(43,122,105,151),(44,131,106,160),(45,140,107,149),(46,129,108,158),(47,138,109,147),(48,127,110,156),(49,136,111,145),(50,125,112,154),(51,134,113,143),(52,123,114,152),(53,132,115,141),(54,121,116,150),(55,130,117,159),(56,139,118,148),(57,128,119,157),(58,137,120,146),(59,126,101,155),(60,135,102,144)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | D10 | C4×D5 | D4⋊2D5 |
| kernel | C2×C23.11D10 | C23.11D10 | C2×C4×Dic5 | C2×C10.D4 | C2×C23.D5 | C10×C22⋊C4 | C23×Dic5 | C22×Dic5 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 8 | 8 | 4 | 2 | 16 | 8 |
Matrix representation of C2×C23.11D10 ►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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 40 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 |
| 0 | 13 | 22 | 0 | 0 |
| 0 | 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 9 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 20 | 0 | 0 |
| 0 | 20 | 38 | 0 | 0 |
| 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 40 | 1 |
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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,40],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,13,0,0,0,19,22,0,0,0,0,0,9,9,0,0,0,23,32],[1,0,0,0,0,0,3,20,0,0,0,20,38,0,0,0,0,0,40,40,0,0,0,2,1] >;
C2×C23.11D10 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{11}D_{10} % in TeX
G:=Group("C2xC2^3.11D10"); // GroupNames label
G:=SmallGroup(320,1152);
// by ID
G=gap.SmallGroup(320,1152);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,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,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations