direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.10D4, (C2×C20)⋊25D4, C23.9(C5×D4), C24.6(C2×C10), C10.92C22≀C2, (C22×D4).3C10, (C22×C10).28D4, C22.70(D4×C10), (C23×C10).6C22, C10.138(C4⋊D4), C2.C42⋊11C10, C10.68(C4.4D4), C23.77(C22×C10), (C22×C10).458C23, (C22×C20).402C22, C10.90(C22.D4), (C2×C4)⋊3(C5×D4), (C2×C4⋊C4)⋊5C10, (C10×C4⋊C4)⋊32C2, (D4×C2×C10).15C2, C2.7(C5×C4⋊D4), (C2×C22⋊C4)⋊7C10, (C10×C22⋊C4)⋊8C2, C2.6(C5×C22≀C2), C2.6(C5×C4.4D4), (C2×C10).610(C2×D4), (C22×C4).6(C2×C10), C22.37(C5×C4○D4), (C2×C10).218(C4○D4), C2.6(C5×C22.D4), (C5×C2.C42)⋊27C2, SmallGroup(320,895)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.10D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce-1 >
Subgroups: 458 in 238 conjugacy classes, 78 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C23.10D4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C23×C10, C5×C2.C42, C10×C22⋊C4, C10×C4⋊C4, D4×C2×C10, C5×C23.10D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C5×D4, C22×C10, C23.10D4, D4×C10, C5×C4○D4, C5×C22≀C2, C5×C4⋊D4, C5×C22.D4, C5×C4.4D4, C5×C23.10D4
►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)
(1 105)(2 101)(3 102)(4 103)(5 104)(6 64)(7 65)(8 61)(9 62)(10 63)(11 125)(12 121)(13 122)(14 123)(15 124)(16 57)(17 58)(18 59)(19 60)(20 56)(21 79)(22 80)(23 76)(24 77)(25 78)(26 130)(27 126)(28 127)(29 128)(30 129)(31 72)(32 73)(33 74)(34 75)(35 71)(36 150)(37 146)(38 147)(39 148)(40 149)(41 135)(42 131)(43 132)(44 133)(45 134)(46 112)(47 113)(48 114)(49 115)(50 111)(51 98)(52 99)(53 100)(54 96)(55 97)(66 117)(67 118)(68 119)(69 120)(70 116)(81 145)(82 141)(83 142)(84 143)(85 144)(86 152)(87 153)(88 154)(89 155)(90 151)(91 138)(92 139)(93 140)(94 136)(95 137)(106 157)(107 158)(108 159)(109 160)(110 156)
(1 45)(2 41)(3 42)(4 43)(5 44)(6 140)(7 136)(8 137)(9 138)(10 139)(11 53)(12 54)(13 55)(14 51)(15 52)(16 141)(17 142)(18 143)(19 144)(20 145)(21 155)(22 151)(23 152)(24 153)(25 154)(26 69)(27 70)(28 66)(29 67)(30 68)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(56 81)(57 82)(58 83)(59 84)(60 85)(61 95)(62 91)(63 92)(64 93)(65 94)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 23)(7 24)(8 25)(9 21)(10 22)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 34)(18 35)(19 31)(20 32)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 113)(97 114)(98 115)(99 111)(100 112)(101 118)(102 119)(103 120)(104 116)(105 117)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 153)(137 154)(138 155)(139 151)(140 152)(141 158)(142 159)(143 160)(144 156)(145 157)
(1 73 54 77)(2 74 55 78)(3 75 51 79)(4 71 52 80)(5 72 53 76)(6 133 19 125)(7 134 20 121)(8 135 16 122)(9 131 17 123)(10 132 18 124)(11 86 44 110)(12 87 45 106)(13 88 41 107)(14 89 42 108)(15 90 43 109)(21 129 34 148)(22 130 35 149)(23 126 31 150)(24 127 32 146)(25 128 33 147)(26 84 40 92)(27 85 36 93)(28 81 37 94)(29 82 38 95)(30 83 39 91)(46 64 70 60)(47 65 66 56)(48 61 67 57)(49 62 68 58)(50 63 69 59)(96 136 105 145)(97 137 101 141)(98 138 102 142)(99 139 103 143)(100 140 104 144)(111 151 120 160)(112 152 116 156)(113 153 117 157)(114 154 118 158)(115 155 119 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 25)(17 21)(18 22)(19 23)(20 24)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 87)(57 88)(58 89)(59 90)(60 86)(61 107)(62 108)(63 109)(64 110)(65 106)(71 92)(72 93)(73 94)(74 95)(75 91)(76 85)(77 81)(78 82)(79 83)(80 84)(96 146)(97 147)(98 148)(99 149)(100 150)(101 128)(102 129)(103 130)(104 126)(105 127)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)(136 157)(137 158)(138 159)(139 160)(140 156)(141 154)(142 155)(143 151)(144 152)(145 153)
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), (1,105)(2,101)(3,102)(4,103)(5,104)(6,64)(7,65)(8,61)(9,62)(10,63)(11,125)(12,121)(13,122)(14,123)(15,124)(16,57)(17,58)(18,59)(19,60)(20,56)(21,79)(22,80)(23,76)(24,77)(25,78)(26,130)(27,126)(28,127)(29,128)(30,129)(31,72)(32,73)(33,74)(34,75)(35,71)(36,150)(37,146)(38,147)(39,148)(40,149)(41,135)(42,131)(43,132)(44,133)(45,134)(46,112)(47,113)(48,114)(49,115)(50,111)(51,98)(52,99)(53,100)(54,96)(55,97)(66,117)(67,118)(68,119)(69,120)(70,116)(81,145)(82,141)(83,142)(84,143)(85,144)(86,152)(87,153)(88,154)(89,155)(90,151)(91,138)(92,139)(93,140)(94,136)(95,137)(106,157)(107,158)(108,159)(109,160)(110,156), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,73,54,77)(2,74,55,78)(3,75,51,79)(4,71,52,80)(5,72,53,76)(6,133,19,125)(7,134,20,121)(8,135,16,122)(9,131,17,123)(10,132,18,124)(11,86,44,110)(12,87,45,106)(13,88,41,107)(14,89,42,108)(15,90,43,109)(21,129,34,148)(22,130,35,149)(23,126,31,150)(24,127,32,146)(25,128,33,147)(26,84,40,92)(27,85,36,93)(28,81,37,94)(29,82,38,95)(30,83,39,91)(46,64,70,60)(47,65,66,56)(48,61,67,57)(49,62,68,58)(50,63,69,59)(96,136,105,145)(97,137,101,141)(98,138,102,142)(99,139,103,143)(100,140,104,144)(111,151,120,160)(112,152,116,156)(113,153,117,157)(114,154,118,158)(115,155,119,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,25)(17,21)(18,22)(19,23)(20,24)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,87)(57,88)(58,89)(59,90)(60,86)(61,107)(62,108)(63,109)(64,110)(65,106)(71,92)(72,93)(73,94)(74,95)(75,91)(76,85)(77,81)(78,82)(79,83)(80,84)(96,146)(97,147)(98,148)(99,149)(100,150)(101,128)(102,129)(103,130)(104,126)(105,127)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132)(136,157)(137,158)(138,159)(139,160)(140,156)(141,154)(142,155)(143,151)(144,152)(145,153)>;
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), (1,105)(2,101)(3,102)(4,103)(5,104)(6,64)(7,65)(8,61)(9,62)(10,63)(11,125)(12,121)(13,122)(14,123)(15,124)(16,57)(17,58)(18,59)(19,60)(20,56)(21,79)(22,80)(23,76)(24,77)(25,78)(26,130)(27,126)(28,127)(29,128)(30,129)(31,72)(32,73)(33,74)(34,75)(35,71)(36,150)(37,146)(38,147)(39,148)(40,149)(41,135)(42,131)(43,132)(44,133)(45,134)(46,112)(47,113)(48,114)(49,115)(50,111)(51,98)(52,99)(53,100)(54,96)(55,97)(66,117)(67,118)(68,119)(69,120)(70,116)(81,145)(82,141)(83,142)(84,143)(85,144)(86,152)(87,153)(88,154)(89,155)(90,151)(91,138)(92,139)(93,140)(94,136)(95,137)(106,157)(107,158)(108,159)(109,160)(110,156), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,73,54,77)(2,74,55,78)(3,75,51,79)(4,71,52,80)(5,72,53,76)(6,133,19,125)(7,134,20,121)(8,135,16,122)(9,131,17,123)(10,132,18,124)(11,86,44,110)(12,87,45,106)(13,88,41,107)(14,89,42,108)(15,90,43,109)(21,129,34,148)(22,130,35,149)(23,126,31,150)(24,127,32,146)(25,128,33,147)(26,84,40,92)(27,85,36,93)(28,81,37,94)(29,82,38,95)(30,83,39,91)(46,64,70,60)(47,65,66,56)(48,61,67,57)(49,62,68,58)(50,63,69,59)(96,136,105,145)(97,137,101,141)(98,138,102,142)(99,139,103,143)(100,140,104,144)(111,151,120,160)(112,152,116,156)(113,153,117,157)(114,154,118,158)(115,155,119,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,25)(17,21)(18,22)(19,23)(20,24)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,87)(57,88)(58,89)(59,90)(60,86)(61,107)(62,108)(63,109)(64,110)(65,106)(71,92)(72,93)(73,94)(74,95)(75,91)(76,85)(77,81)(78,82)(79,83)(80,84)(96,146)(97,147)(98,148)(99,149)(100,150)(101,128)(102,129)(103,130)(104,126)(105,127)(111,124)(112,125)(113,121)(114,122)(115,123)(116,133)(117,134)(118,135)(119,131)(120,132)(136,157)(137,158)(138,159)(139,160)(140,156)(141,154)(142,155)(143,151)(144,152)(145,153) );
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)], [(1,105),(2,101),(3,102),(4,103),(5,104),(6,64),(7,65),(8,61),(9,62),(10,63),(11,125),(12,121),(13,122),(14,123),(15,124),(16,57),(17,58),(18,59),(19,60),(20,56),(21,79),(22,80),(23,76),(24,77),(25,78),(26,130),(27,126),(28,127),(29,128),(30,129),(31,72),(32,73),(33,74),(34,75),(35,71),(36,150),(37,146),(38,147),(39,148),(40,149),(41,135),(42,131),(43,132),(44,133),(45,134),(46,112),(47,113),(48,114),(49,115),(50,111),(51,98),(52,99),(53,100),(54,96),(55,97),(66,117),(67,118),(68,119),(69,120),(70,116),(81,145),(82,141),(83,142),(84,143),(85,144),(86,152),(87,153),(88,154),(89,155),(90,151),(91,138),(92,139),(93,140),(94,136),(95,137),(106,157),(107,158),(108,159),(109,160),(110,156)], [(1,45),(2,41),(3,42),(4,43),(5,44),(6,140),(7,136),(8,137),(9,138),(10,139),(11,53),(12,54),(13,55),(14,51),(15,52),(16,141),(17,142),(18,143),(19,144),(20,145),(21,155),(22,151),(23,152),(24,153),(25,154),(26,69),(27,70),(28,66),(29,67),(30,68),(31,156),(32,157),(33,158),(34,159),(35,160),(36,46),(37,47),(38,48),(39,49),(40,50),(56,81),(57,82),(58,83),(59,84),(60,85),(61,95),(62,91),(63,92),(64,93),(65,94),(71,109),(72,110),(73,106),(74,107),(75,108),(76,86),(77,87),(78,88),(79,89),(80,90),(96,121),(97,122),(98,123),(99,124),(100,125),(101,135),(102,131),(103,132),(104,133),(105,134),(111,149),(112,150),(113,146),(114,147),(115,148),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,23),(7,24),(8,25),(9,21),(10,22),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,34),(18,35),(19,31),(20,32),(26,43),(27,44),(28,45),(29,41),(30,42),(46,53),(47,54),(48,55),(49,51),(50,52),(56,73),(57,74),(58,75),(59,71),(60,72),(61,78),(62,79),(63,80),(64,76),(65,77),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,113),(97,114),(98,115),(99,111),(100,112),(101,118),(102,119),(103,120),(104,116),(105,117),(121,146),(122,147),(123,148),(124,149),(125,150),(126,133),(127,134),(128,135),(129,131),(130,132),(136,153),(137,154),(138,155),(139,151),(140,152),(141,158),(142,159),(143,160),(144,156),(145,157)], [(1,73,54,77),(2,74,55,78),(3,75,51,79),(4,71,52,80),(5,72,53,76),(6,133,19,125),(7,134,20,121),(8,135,16,122),(9,131,17,123),(10,132,18,124),(11,86,44,110),(12,87,45,106),(13,88,41,107),(14,89,42,108),(15,90,43,109),(21,129,34,148),(22,130,35,149),(23,126,31,150),(24,127,32,146),(25,128,33,147),(26,84,40,92),(27,85,36,93),(28,81,37,94),(29,82,38,95),(30,83,39,91),(46,64,70,60),(47,65,66,56),(48,61,67,57),(49,62,68,58),(50,63,69,59),(96,136,105,145),(97,137,101,141),(98,138,102,142),(99,139,103,143),(100,140,104,144),(111,151,120,160),(112,152,116,156),(113,153,117,157),(114,154,118,158),(115,155,119,159)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,25),(17,21),(18,22),(19,23),(20,24),(26,43),(27,44),(28,45),(29,41),(30,42),(46,53),(47,54),(48,55),(49,51),(50,52),(56,87),(57,88),(58,89),(59,90),(60,86),(61,107),(62,108),(63,109),(64,110),(65,106),(71,92),(72,93),(73,94),(74,95),(75,91),(76,85),(77,81),(78,82),(79,83),(80,84),(96,146),(97,147),(98,148),(99,149),(100,150),(101,128),(102,129),(103,130),(104,126),(105,127),(111,124),(112,125),(113,121),(114,122),(115,123),(116,133),(117,134),(118,135),(119,131),(120,132),(136,157),(137,158),(138,159),(139,160),(140,156),(141,154),(142,155),(143,151),(144,152),(145,153)]])
110 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4J | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AN |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C23.10D4 | C5×C2.C42 | C10×C22⋊C4 | C10×C4⋊C4 | D4×C2×C10 | C23.10D4 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 4 | 4 | 6 | 16 | 16 | 24 |
Matrix representation of C5×C23.10D4 ►in GL6(𝔽41)
| 37 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 40 | 31 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 37 |
| 0 | 0 | 0 | 0 | 33 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 31 | 11 | 0 | 0 | 0 | 0 |
| 2 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 10 | 32 |
| 40 | 31 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 33 | 40 |
G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,31,1,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,25,33,0,0,0,0,37,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,2,0,0,0,0,11,10,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,9,10,0,0,0,0,0,32],[40,0,0,0,0,0,31,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,33,0,0,0,0,0,40] >;
C5×C23.10D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{10}D_4 % in TeX
G:=Group("C5xC2^3.10D4"); // GroupNames label
G:=SmallGroup(320,895);
// by ID
G=gap.SmallGroup(320,895);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1731,226]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,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^-1>;
// generators/relations