direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.23D4, (C2×D4)⋊3C20, (C2×C20)⋊36D4, C2.5(D4×C20), (D4×C10)⋊27C4, (C23×C4)⋊1C10, C23⋊2(C2×C20), (C23×C20)⋊2C2, C10.136(C4×D4), C23.22(C5×D4), C10.89C22≀C2, C24.27(C2×C10), (C22×D4).1C10, C22.35(D4×C10), C2.C42⋊8C10, (C22×C10).156D4, C10.133(C4⋊D4), C23.59(C22×C10), C22.35(C22×C20), (C23×C10).87C22, (C22×C10).450C23, (C22×C20).494C22, C10.87(C22.D4), (C2×C4)⋊9(C5×D4), (C2×C4)⋊3(C2×C20), (C2×C20)⋊37(C2×C4), (D4×C2×C10).13C2, C2.2(C5×C4⋊D4), (C2×C22⋊C4)⋊2C10, (C10×C22⋊C4)⋊6C2, C2.3(C5×C22≀C2), C2.7(C10×C22⋊C4), C22⋊1(C5×C22⋊C4), (C2×C10)⋊7(C22⋊C4), (C22×C10)⋊11(C2×C4), (C2×C10).602(C2×D4), C22.20(C5×C4○D4), C10.135(C2×C22⋊C4), (C22×C4).87(C2×C10), (C2×C10).210(C4○D4), C2.3(C5×C22.D4), (C5×C2.C42)⋊24C2, (C2×C10).323(C22×C4), SmallGroup(320,887)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.23D4
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, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >
Subgroups: 498 in 286 conjugacy classes, 106 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, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C23.23D4, C5×C22⋊C4, C22×C20, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C5×C2.C42, C10×C22⋊C4, C10×C22⋊C4, C23×C20, D4×C2×C10, C5×C23.23D4
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, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×C20, C5×D4, C22×C10, C23.23D4, C5×C22⋊C4, C22×C20, D4×C10, C5×C4○D4, C10×C22⋊C4, D4×C20, C5×C22≀C2, C5×C4⋊D4, C5×C22.D4, C5×C23.23D4
►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 81)(2 82)(3 83)(4 84)(5 85)(6 125)(7 121)(8 122)(9 123)(10 124)(11 63)(12 64)(13 65)(14 61)(15 62)(16 146)(17 147)(18 148)(19 149)(20 150)(21 132)(22 133)(23 134)(24 135)(25 131)(26 59)(27 60)(28 56)(29 57)(30 58)(31 126)(32 127)(33 128)(34 129)(35 130)(36 76)(37 77)(38 78)(39 79)(40 80)(41 72)(42 73)(43 74)(44 75)(45 71)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(66 106)(67 107)(68 108)(69 109)(70 110)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 20)(7 16)(8 17)(9 18)(10 19)(11 41)(12 42)(13 43)(14 44)(15 45)(21 35)(22 31)(23 32)(24 33)(25 34)(26 39)(27 40)(28 36)(29 37)(30 38)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 116)(97 117)(98 118)(99 119)(100 120)(101 115)(102 111)(103 112)(104 113)(105 114)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 156)(137 157)(138 158)(139 159)(140 160)(141 155)(142 151)(143 152)(144 153)(145 154)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 136)(7 137)(8 138)(9 139)(10 140)(11 53)(12 54)(13 55)(14 51)(15 52)(16 157)(17 158)(18 159)(19 160)(20 156)(21 142)(22 143)(23 144)(24 145)(25 141)(31 152)(32 153)(33 154)(34 155)(35 151)(36 70)(37 66)(38 67)(39 68)(40 69)(41 46)(42 47)(43 48)(44 49)(45 50)(56 85)(57 81)(58 82)(59 83)(60 84)(61 91)(62 92)(63 93)(64 94)(65 95)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)(96 125)(97 121)(98 122)(99 123)(100 124)(101 131)(102 132)(103 133)(104 134)(105 135)(111 130)(112 126)(113 127)(114 128)(115 129)(116 150)(117 146)(118 147)(119 148)(120 149)
(1 134 54 121)(2 135 55 122)(3 131 51 123)(4 132 52 124)(5 133 53 125)(6 85 22 93)(7 81 23 94)(8 82 24 95)(9 83 25 91)(10 84 21 92)(11 96 28 103)(12 97 29 104)(13 98 30 105)(14 99 26 101)(15 100 27 102)(16 106 32 87)(17 107 33 88)(18 108 34 89)(19 109 35 90)(20 110 31 86)(36 112 41 116)(37 113 42 117)(38 114 43 118)(39 115 44 119)(40 111 45 120)(46 150 70 126)(47 146 66 127)(48 147 67 128)(49 148 68 129)(50 149 69 130)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 141 61 139)(60 142 62 140)(71 160 80 151)(72 156 76 152)(73 157 77 153)(74 158 78 154)(75 159 79 155)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 20)(7 16)(8 17)(9 18)(10 19)(21 35)(22 31)(23 32)(24 33)(25 34)(26 51)(27 52)(28 53)(29 54)(30 55)(36 46)(37 47)(38 48)(39 49)(40 50)(41 70)(42 66)(43 67)(44 68)(45 69)(56 86)(57 87)(58 88)(59 89)(60 90)(61 108)(62 109)(63 110)(64 106)(65 107)(71 84)(72 85)(73 81)(74 82)(75 83)(76 93)(77 94)(78 95)(79 91)(80 92)(136 156)(137 157)(138 158)(139 159)(140 160)(141 155)(142 151)(143 152)(144 153)(145 154)
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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,125)(7,121)(8,122)(9,123)(10,124)(11,63)(12,64)(13,65)(14,61)(15,62)(16,146)(17,147)(18,148)(19,149)(20,150)(21,132)(22,133)(23,134)(24,135)(25,131)(26,59)(27,60)(28,56)(29,57)(30,58)(31,126)(32,127)(33,128)(34,129)(35,130)(36,76)(37,77)(38,78)(39,79)(40,80)(41,72)(42,73)(43,74)(44,75)(45,71)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(66,106)(67,107)(68,108)(69,109)(70,110)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,66)(2,67)(3,68)(4,69)(5,70)(6,20)(7,16)(8,17)(9,18)(10,19)(11,41)(12,42)(13,43)(14,44)(15,45)(21,35)(22,31)(23,32)(24,33)(25,34)(26,39)(27,40)(28,36)(29,37)(30,38)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,116)(97,117)(98,118)(99,119)(100,120)(101,115)(102,111)(103,112)(104,113)(105,114)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154), (1,29)(2,30)(3,26)(4,27)(5,28)(6,136)(7,137)(8,138)(9,139)(10,140)(11,53)(12,54)(13,55)(14,51)(15,52)(16,157)(17,158)(18,159)(19,160)(20,156)(21,142)(22,143)(23,144)(24,145)(25,141)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,46)(42,47)(43,48)(44,49)(45,50)(56,85)(57,81)(58,82)(59,83)(60,84)(61,91)(62,92)(63,93)(64,94)(65,95)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,125)(97,121)(98,122)(99,123)(100,124)(101,131)(102,132)(103,133)(104,134)(105,135)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,134,54,121)(2,135,55,122)(3,131,51,123)(4,132,52,124)(5,133,53,125)(6,85,22,93)(7,81,23,94)(8,82,24,95)(9,83,25,91)(10,84,21,92)(11,96,28,103)(12,97,29,104)(13,98,30,105)(14,99,26,101)(15,100,27,102)(16,106,32,87)(17,107,33,88)(18,108,34,89)(19,109,35,90)(20,110,31,86)(36,112,41,116)(37,113,42,117)(38,114,43,118)(39,115,44,119)(40,111,45,120)(46,150,70,126)(47,146,66,127)(48,147,67,128)(49,148,68,129)(50,149,69,130)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,141,61,139)(60,142,62,140)(71,160,80,151)(72,156,76,152)(73,157,77,153)(74,158,78,154)(75,159,79,155), (1,12)(2,13)(3,14)(4,15)(5,11)(6,20)(7,16)(8,17)(9,18)(10,19)(21,35)(22,31)(23,32)(24,33)(25,34)(26,51)(27,52)(28,53)(29,54)(30,55)(36,46)(37,47)(38,48)(39,49)(40,50)(41,70)(42,66)(43,67)(44,68)(45,69)(56,86)(57,87)(58,88)(59,89)(60,90)(61,108)(62,109)(63,110)(64,106)(65,107)(71,84)(72,85)(73,81)(74,82)(75,83)(76,93)(77,94)(78,95)(79,91)(80,92)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154)>;
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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,125)(7,121)(8,122)(9,123)(10,124)(11,63)(12,64)(13,65)(14,61)(15,62)(16,146)(17,147)(18,148)(19,149)(20,150)(21,132)(22,133)(23,134)(24,135)(25,131)(26,59)(27,60)(28,56)(29,57)(30,58)(31,126)(32,127)(33,128)(34,129)(35,130)(36,76)(37,77)(38,78)(39,79)(40,80)(41,72)(42,73)(43,74)(44,75)(45,71)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(66,106)(67,107)(68,108)(69,109)(70,110)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,66)(2,67)(3,68)(4,69)(5,70)(6,20)(7,16)(8,17)(9,18)(10,19)(11,41)(12,42)(13,43)(14,44)(15,45)(21,35)(22,31)(23,32)(24,33)(25,34)(26,39)(27,40)(28,36)(29,37)(30,38)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,116)(97,117)(98,118)(99,119)(100,120)(101,115)(102,111)(103,112)(104,113)(105,114)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154), (1,29)(2,30)(3,26)(4,27)(5,28)(6,136)(7,137)(8,138)(9,139)(10,140)(11,53)(12,54)(13,55)(14,51)(15,52)(16,157)(17,158)(18,159)(19,160)(20,156)(21,142)(22,143)(23,144)(24,145)(25,141)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,46)(42,47)(43,48)(44,49)(45,50)(56,85)(57,81)(58,82)(59,83)(60,84)(61,91)(62,92)(63,93)(64,94)(65,95)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,125)(97,121)(98,122)(99,123)(100,124)(101,131)(102,132)(103,133)(104,134)(105,135)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,134,54,121)(2,135,55,122)(3,131,51,123)(4,132,52,124)(5,133,53,125)(6,85,22,93)(7,81,23,94)(8,82,24,95)(9,83,25,91)(10,84,21,92)(11,96,28,103)(12,97,29,104)(13,98,30,105)(14,99,26,101)(15,100,27,102)(16,106,32,87)(17,107,33,88)(18,108,34,89)(19,109,35,90)(20,110,31,86)(36,112,41,116)(37,113,42,117)(38,114,43,118)(39,115,44,119)(40,111,45,120)(46,150,70,126)(47,146,66,127)(48,147,67,128)(49,148,68,129)(50,149,69,130)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,141,61,139)(60,142,62,140)(71,160,80,151)(72,156,76,152)(73,157,77,153)(74,158,78,154)(75,159,79,155), (1,12)(2,13)(3,14)(4,15)(5,11)(6,20)(7,16)(8,17)(9,18)(10,19)(21,35)(22,31)(23,32)(24,33)(25,34)(26,51)(27,52)(28,53)(29,54)(30,55)(36,46)(37,47)(38,48)(39,49)(40,50)(41,70)(42,66)(43,67)(44,68)(45,69)(56,86)(57,87)(58,88)(59,89)(60,90)(61,108)(62,109)(63,110)(64,106)(65,107)(71,84)(72,85)(73,81)(74,82)(75,83)(76,93)(77,94)(78,95)(79,91)(80,92)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154) );
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)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,125),(7,121),(8,122),(9,123),(10,124),(11,63),(12,64),(13,65),(14,61),(15,62),(16,146),(17,147),(18,148),(19,149),(20,150),(21,132),(22,133),(23,134),(24,135),(25,131),(26,59),(27,60),(28,56),(29,57),(30,58),(31,126),(32,127),(33,128),(34,129),(35,130),(36,76),(37,77),(38,78),(39,79),(40,80),(41,72),(42,73),(43,74),(44,75),(45,71),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(66,106),(67,107),(68,108),(69,109),(70,110),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,20),(7,16),(8,17),(9,18),(10,19),(11,41),(12,42),(13,43),(14,44),(15,45),(21,35),(22,31),(23,32),(24,33),(25,34),(26,39),(27,40),(28,36),(29,37),(30,38),(46,53),(47,54),(48,55),(49,51),(50,52),(56,76),(57,77),(58,78),(59,79),(60,80),(61,75),(62,71),(63,72),(64,73),(65,74),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,116),(97,117),(98,118),(99,119),(100,120),(101,115),(102,111),(103,112),(104,113),(105,114),(121,146),(122,147),(123,148),(124,149),(125,150),(126,133),(127,134),(128,135),(129,131),(130,132),(136,156),(137,157),(138,158),(139,159),(140,160),(141,155),(142,151),(143,152),(144,153),(145,154)], [(1,29),(2,30),(3,26),(4,27),(5,28),(6,136),(7,137),(8,138),(9,139),(10,140),(11,53),(12,54),(13,55),(14,51),(15,52),(16,157),(17,158),(18,159),(19,160),(20,156),(21,142),(22,143),(23,144),(24,145),(25,141),(31,152),(32,153),(33,154),(34,155),(35,151),(36,70),(37,66),(38,67),(39,68),(40,69),(41,46),(42,47),(43,48),(44,49),(45,50),(56,85),(57,81),(58,82),(59,83),(60,84),(61,91),(62,92),(63,93),(64,94),(65,95),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109),(96,125),(97,121),(98,122),(99,123),(100,124),(101,131),(102,132),(103,133),(104,134),(105,135),(111,130),(112,126),(113,127),(114,128),(115,129),(116,150),(117,146),(118,147),(119,148),(120,149)], [(1,134,54,121),(2,135,55,122),(3,131,51,123),(4,132,52,124),(5,133,53,125),(6,85,22,93),(7,81,23,94),(8,82,24,95),(9,83,25,91),(10,84,21,92),(11,96,28,103),(12,97,29,104),(13,98,30,105),(14,99,26,101),(15,100,27,102),(16,106,32,87),(17,107,33,88),(18,108,34,89),(19,109,35,90),(20,110,31,86),(36,112,41,116),(37,113,42,117),(38,114,43,118),(39,115,44,119),(40,111,45,120),(46,150,70,126),(47,146,66,127),(48,147,67,128),(49,148,68,129),(50,149,69,130),(56,143,63,136),(57,144,64,137),(58,145,65,138),(59,141,61,139),(60,142,62,140),(71,160,80,151),(72,156,76,152),(73,157,77,153),(74,158,78,154),(75,159,79,155)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,20),(7,16),(8,17),(9,18),(10,19),(21,35),(22,31),(23,32),(24,33),(25,34),(26,51),(27,52),(28,53),(29,54),(30,55),(36,46),(37,47),(38,48),(39,49),(40,50),(41,70),(42,66),(43,67),(44,68),(45,69),(56,86),(57,87),(58,88),(59,89),(60,90),(61,108),(62,109),(63,110),(64,106),(65,107),(71,84),(72,85),(73,81),(74,82),(75,83),(76,93),(77,94),(78,95),(79,91),(80,92),(136,156),(137,157),(138,158),(139,159),(140,160),(141,155),(142,151),(143,152),(144,153),(145,154)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10AZ | 20A | ··· | 20AF | 20AG | ··· | 20BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C23.23D4 | C5×C2.C42 | C10×C22⋊C4 | C23×C20 | D4×C2×C10 | D4×C10 | C23.23D4 | C2.C42 | C2×C22⋊C4 | C23×C4 | C22×D4 | C2×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 1 | 1 | 8 | 4 | 8 | 12 | 4 | 4 | 32 | 4 | 4 | 4 | 16 | 16 | 16 |
Matrix representation of C5×C23.23D4 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[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],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,40,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1] >;
C5×C23.23D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{23}D_4 % in TeX
G:=Group("C5xC2^3.23D4"); // GroupNames label
G:=SmallGroup(320,887);
// by ID
G=gap.SmallGroup(320,887);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766]);
// 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,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
// generators/relations