direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.24D4, C4○D4⋊1C20, (C22×C40)⋊8C2, (C22×C8)⋊4C10, D4.5(C2×C20), C4.53(D4×C10), Q8.5(C2×C20), C20.460(C2×D4), (C2×C20).518D4, C4.3(C22×C20), D4⋊C4⋊20C10, C23.23(C5×D4), C42⋊C2⋊2C10, Q8⋊C4⋊20C10, C22.43(D4×C10), C10.114(C4○D8), (C2×C40).358C22, C20.207(C22×C4), (C2×C20).892C23, (C22×C10).127D4, C20.163(C22⋊C4), (D4×C10).287C22, (Q8×C10).251C22, (C22×C20).583C22, C2.1(C5×C4○D8), (C5×C4○D4)⋊13C4, C4⋊C4.38(C2×C10), (C2×C8).61(C2×C10), (C2×C4).49(C2×C20), (C2×C4○D4).4C10, (C5×D4).41(C2×C4), (C2×C4).122(C5×D4), C4.32(C5×C22⋊C4), (C5×Q8).44(C2×C4), (C5×D4⋊C4)⋊43C2, (C2×C20).443(C2×C4), (C5×Q8⋊C4)⋊43C2, (C10×C4○D4).18C2, (C2×D4).45(C2×C10), (C2×C10).619(C2×D4), C2.19(C10×C22⋊C4), (C2×Q8).36(C2×C10), C22.4(C5×C22⋊C4), (C5×C42⋊C2)⋊23C2, (C5×C4⋊C4).359C22, C10.148(C2×C22⋊C4), (C2×C4).67(C22×C10), (C2×C10).95(C22⋊C4), (C22×C4).112(C2×C10), SmallGroup(320,917)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.24D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 258 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C23.24D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×D4⋊C4, C5×Q8⋊C4, C5×C42⋊C2, C22×C40, C10×C4○D4, C5×C23.24D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C4○D8, C2×C20, C5×D4, C22×C10, C23.24D4, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C4○D8, C5×C23.24D4
►On 160 points
Generators in S160
(1 98 23 90 15)(2 99 24 91 16)(3 100 17 92 9)(4 101 18 93 10)(5 102 19 94 11)(6 103 20 95 12)(7 104 21 96 13)(8 97 22 89 14)(25 41 116 33 108)(26 42 117 34 109)(27 43 118 35 110)(28 44 119 36 111)(29 45 120 37 112)(30 46 113 38 105)(31 47 114 39 106)(32 48 115 40 107)(49 124 140 57 132)(50 125 141 58 133)(51 126 142 59 134)(52 127 143 60 135)(53 128 144 61 136)(54 121 137 62 129)(55 122 138 63 130)(56 123 139 64 131)(65 82 156 73 148)(66 83 157 74 149)(67 84 158 75 150)(68 85 159 76 151)(69 86 160 77 152)(70 87 153 78 145)(71 88 154 79 146)(72 81 155 80 147)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 145)(16 146)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 153)(24 154)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 135)(34 136)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 143)(42 144)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(73 93)(74 94)(75 95)(76 96)(77 89)(78 90)(79 91)(80 92)(81 100)(82 101)(83 102)(84 103)(85 104)(86 97)(87 98)(88 99)
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 87)(42 88)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)(97 142)(98 143)(99 144)(100 137)(101 138)(102 139)(103 140)(104 141)(105 150)(106 151)(107 152)(108 145)(109 146)(110 147)(111 148)(112 149)(113 158)(114 159)(115 160)(116 153)(117 154)(118 155)(119 156)(120 157)
(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)
(1 8 127 126)(2 125 128 7)(3 6 121 124)(4 123 122 5)(9 12 54 49)(10 56 55 11)(13 16 50 53)(14 52 51 15)(17 20 62 57)(18 64 63 19)(21 24 58 61)(22 60 59 23)(25 28 70 65)(26 72 71 27)(29 32 66 69)(30 68 67 31)(33 36 78 73)(34 80 79 35)(37 40 74 77)(38 76 75 39)(41 44 87 82)(42 81 88 43)(45 48 83 86)(46 85 84 47)(89 135 134 90)(91 133 136 96)(92 95 129 132)(93 131 130 94)(97 143 142 98)(99 141 144 104)(100 103 137 140)(101 139 138 102)(105 151 150 106)(107 149 152 112)(108 111 145 148)(109 147 146 110)(113 159 158 114)(115 157 160 120)(116 119 153 156)(117 155 154 118)
G:=sub<Sym(160)| (1,98,23,90,15)(2,99,24,91,16)(3,100,17,92,9)(4,101,18,93,10)(5,102,19,94,11)(6,103,20,95,12)(7,104,21,96,13)(8,97,22,89,14)(25,41,116,33,108)(26,42,117,34,109)(27,43,118,35,110)(28,44,119,36,111)(29,45,120,37,112)(30,46,113,38,105)(31,47,114,39,106)(32,48,115,40,107)(49,124,140,57,132)(50,125,141,58,133)(51,126,142,59,134)(52,127,143,60,135)(53,128,144,61,136)(54,121,137,62,129)(55,122,138,63,130)(56,123,139,64,131)(65,82,156,73,148)(66,83,157,74,149)(67,84,158,75,150)(68,85,159,76,151)(69,86,160,77,152)(70,87,153,78,145)(71,88,154,79,146)(72,81,155,80,147), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,145)(16,146)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,153)(24,154)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,135)(34,136)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,143)(42,144)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92)(81,100)(82,101)(83,102)(84,103)(85,104)(86,97)(87,98)(88,99), (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133)(97,142)(98,143)(99,144)(100,137)(101,138)(102,139)(103,140)(104,141)(105,150)(106,151)(107,152)(108,145)(109,146)(110,147)(111,148)(112,149)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157), (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), (1,8,127,126)(2,125,128,7)(3,6,121,124)(4,123,122,5)(9,12,54,49)(10,56,55,11)(13,16,50,53)(14,52,51,15)(17,20,62,57)(18,64,63,19)(21,24,58,61)(22,60,59,23)(25,28,70,65)(26,72,71,27)(29,32,66,69)(30,68,67,31)(33,36,78,73)(34,80,79,35)(37,40,74,77)(38,76,75,39)(41,44,87,82)(42,81,88,43)(45,48,83,86)(46,85,84,47)(89,135,134,90)(91,133,136,96)(92,95,129,132)(93,131,130,94)(97,143,142,98)(99,141,144,104)(100,103,137,140)(101,139,138,102)(105,151,150,106)(107,149,152,112)(108,111,145,148)(109,147,146,110)(113,159,158,114)(115,157,160,120)(116,119,153,156)(117,155,154,118)>;
G:=Group( (1,98,23,90,15)(2,99,24,91,16)(3,100,17,92,9)(4,101,18,93,10)(5,102,19,94,11)(6,103,20,95,12)(7,104,21,96,13)(8,97,22,89,14)(25,41,116,33,108)(26,42,117,34,109)(27,43,118,35,110)(28,44,119,36,111)(29,45,120,37,112)(30,46,113,38,105)(31,47,114,39,106)(32,48,115,40,107)(49,124,140,57,132)(50,125,141,58,133)(51,126,142,59,134)(52,127,143,60,135)(53,128,144,61,136)(54,121,137,62,129)(55,122,138,63,130)(56,123,139,64,131)(65,82,156,73,148)(66,83,157,74,149)(67,84,158,75,150)(68,85,159,76,151)(69,86,160,77,152)(70,87,153,78,145)(71,88,154,79,146)(72,81,155,80,147), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,145)(16,146)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,153)(24,154)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,135)(34,136)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,143)(42,144)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92)(81,100)(82,101)(83,102)(84,103)(85,104)(86,97)(87,98)(88,99), (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133)(97,142)(98,143)(99,144)(100,137)(101,138)(102,139)(103,140)(104,141)(105,150)(106,151)(107,152)(108,145)(109,146)(110,147)(111,148)(112,149)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157), (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), (1,8,127,126)(2,125,128,7)(3,6,121,124)(4,123,122,5)(9,12,54,49)(10,56,55,11)(13,16,50,53)(14,52,51,15)(17,20,62,57)(18,64,63,19)(21,24,58,61)(22,60,59,23)(25,28,70,65)(26,72,71,27)(29,32,66,69)(30,68,67,31)(33,36,78,73)(34,80,79,35)(37,40,74,77)(38,76,75,39)(41,44,87,82)(42,81,88,43)(45,48,83,86)(46,85,84,47)(89,135,134,90)(91,133,136,96)(92,95,129,132)(93,131,130,94)(97,143,142,98)(99,141,144,104)(100,103,137,140)(101,139,138,102)(105,151,150,106)(107,149,152,112)(108,111,145,148)(109,147,146,110)(113,159,158,114)(115,157,160,120)(116,119,153,156)(117,155,154,118) );
G=PermutationGroup([[(1,98,23,90,15),(2,99,24,91,16),(3,100,17,92,9),(4,101,18,93,10),(5,102,19,94,11),(6,103,20,95,12),(7,104,21,96,13),(8,97,22,89,14),(25,41,116,33,108),(26,42,117,34,109),(27,43,118,35,110),(28,44,119,36,111),(29,45,120,37,112),(30,46,113,38,105),(31,47,114,39,106),(32,48,115,40,107),(49,124,140,57,132),(50,125,141,58,133),(51,126,142,59,134),(52,127,143,60,135),(53,128,144,61,136),(54,121,137,62,129),(55,122,138,63,130),(56,123,139,64,131),(65,82,156,73,148),(66,83,157,74,149),(67,84,158,75,150),(68,85,159,76,151),(69,86,160,77,152),(70,87,153,78,145),(71,88,154,79,146),(72,81,155,80,147)], [(1,70),(2,71),(3,72),(4,65),(5,66),(6,67),(7,68),(8,69),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,145),(16,146),(17,155),(18,156),(19,157),(20,158),(21,159),(22,160),(23,153),(24,154),(25,127),(26,128),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,135),(34,136),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,143),(42,144),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(73,93),(74,94),(75,95),(76,96),(77,89),(78,90),(79,91),(80,92),(81,100),(82,101),(83,102),(84,103),(85,104),(86,97),(87,98),(88,99)], [(1,127),(2,128),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(33,78),(34,79),(35,80),(36,73),(37,74),(38,75),(39,76),(40,77),(41,87),(42,88),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(89,134),(90,135),(91,136),(92,129),(93,130),(94,131),(95,132),(96,133),(97,142),(98,143),(99,144),(100,137),(101,138),(102,139),(103,140),(104,141),(105,150),(106,151),(107,152),(108,145),(109,146),(110,147),(111,148),(112,149),(113,158),(114,159),(115,160),(116,153),(117,154),(118,155),(119,156),(120,157)], [(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)], [(1,8,127,126),(2,125,128,7),(3,6,121,124),(4,123,122,5),(9,12,54,49),(10,56,55,11),(13,16,50,53),(14,52,51,15),(17,20,62,57),(18,64,63,19),(21,24,58,61),(22,60,59,23),(25,28,70,65),(26,72,71,27),(29,32,66,69),(30,68,67,31),(33,36,78,73),(34,80,79,35),(37,40,74,77),(38,76,75,39),(41,44,87,82),(42,81,88,43),(45,48,83,86),(46,85,84,47),(89,135,134,90),(91,133,136,96),(92,95,129,132),(93,131,130,94),(97,143,142,98),(99,141,144,104),(100,103,137,140),(101,139,138,102),(105,151,150,106),(107,149,152,112),(108,111,145,148),(109,147,146,110),(113,159,158,114),(115,157,160,120),(116,119,153,156),(117,155,154,118)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AV | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 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 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D8 |
| kernel | C5×C23.24D4 | C5×D4⋊C4 | C5×Q8⋊C4 | C5×C42⋊C2 | C22×C40 | C10×C4○D4 | C5×C4○D4 | C23.24D4 | D4⋊C4 | Q8⋊C4 | C42⋊C2 | C22×C8 | C2×C4○D4 | C4○D4 | C2×C20 | C22×C10 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 4 | 4 | 32 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of C5×C23.24D4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 32 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 29 | 29 |
| 9 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,0,32,0,0,9,0],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,40,0,0,0,0,29,29,0,0,12,29],[9,0,0,0,0,1,0,0,0,0,29,12,0,0,12,12] >;
C5×C23.24D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{24}D_4 % in TeX
G:=Group("C5xC2^3.24D4"); // GroupNames label
G:=SmallGroup(320,917);
// by ID
G=gap.SmallGroup(320,917);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,856,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,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*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations