direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.2F5, C24.3F5, C23⋊2(C5⋊C8), (C22×C10)⋊4C8, C10⋊2(C22⋊C8), (C23×C10).6C4, C23.63(C2×F5), C10.24(C22×C8), Dic5.116(C2×D4), (C2×Dic5).263D4, (C2×C10).25M4(2), C10.33(C2×M4(2)), C22.53(C22×F5), (C23×Dic5).12C2, (C22×Dic5).36C4, C22.53(C22⋊F5), Dic5.51(C22⋊C4), (C2×Dic5).359C23, C22.10(C22.F5), (C22×Dic5).279C22, C5⋊3(C2×C22⋊C8), C22⋊2(C2×C5⋊C8), (C2×C10)⋊8(C2×C8), (C22×C5⋊C8)⋊7C2, (C2×C5⋊C8)⋊8C22, C2.9(C22×C5⋊C8), C2.5(C2×C22⋊F5), C10.38(C2×C22⋊C4), C2.5(C2×C22.F5), (C2×C10).91(C22×C4), (C22×C10).73(C2×C4), (C2×C10).62(C22⋊C4), (C2×Dic5).195(C2×C4), SmallGroup(320,1135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.2F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 602 in 202 conjugacy classes, 84 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, C22×C4, C24, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C23×C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C2×C22⋊C8, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, C22×Dic5, C22×Dic5, C23×C10, C23.2F5, C22×C5⋊C8, C23×Dic5, C2×C23.2F5
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, F5, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C5⋊C8, C2×F5, C2×C22⋊C8, C2×C5⋊C8, C22.F5, C22⋊F5, C22×F5, C23.2F5, C22×C5⋊C8, C2×C22.F5, C2×C22⋊F5, C2×C23.2F5
►On 160 points
Generators in S160
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 81)(16 82)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(41 127)(42 128)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 153)(72 154)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(97 119)(98 120)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(129 140)(130 141)(131 142)(132 143)(133 144)(134 137)(135 138)(136 139)
(1 48)(2 29)(3 42)(4 31)(5 44)(6 25)(7 46)(8 27)(9 151)(10 88)(11 145)(12 82)(13 147)(14 84)(15 149)(16 86)(17 92)(18 105)(19 94)(20 107)(21 96)(22 109)(23 90)(24 111)(26 128)(28 122)(30 124)(32 126)(33 135)(34 58)(35 129)(36 60)(37 131)(38 62)(39 133)(40 64)(41 123)(43 125)(45 127)(47 121)(49 85)(50 150)(51 87)(52 152)(53 81)(54 146)(55 83)(56 148)(57 142)(59 144)(61 138)(63 140)(65 97)(66 160)(67 99)(68 154)(69 101)(70 156)(71 103)(72 158)(73 112)(74 93)(75 106)(76 95)(77 108)(78 89)(79 110)(80 91)(98 116)(100 118)(102 120)(104 114)(113 157)(115 159)(117 153)(119 155)(130 137)(132 139)(134 141)(136 143)
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 121)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 49)(16 50)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 142)(34 143)(35 144)(36 137)(37 138)(38 139)(39 140)(40 141)(57 135)(58 136)(59 129)(60 130)(61 131)(62 132)(63 133)(64 134)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 109)(90 110)(91 111)(92 112)(93 105)(94 106)(95 107)(96 108)(97 159)(98 160)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)
(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)
(1 159 63 55 75)(2 56 160 76 64)(3 77 49 57 153)(4 58 78 154 50)(5 155 59 51 79)(6 52 156 80 60)(7 73 53 61 157)(8 62 74 158 54)(9 23 126 101 129)(10 102 24 130 127)(11 131 103 128 17)(12 121 132 18 104)(13 19 122 97 133)(14 98 20 134 123)(15 135 99 124 21)(16 125 136 22 100)(25 152 70 91 36)(26 92 145 37 71)(27 38 93 72 146)(28 65 39 147 94)(29 148 66 95 40)(30 96 149 33 67)(31 34 89 68 150)(32 69 35 151 90)(41 84 116 107 141)(42 108 85 142 117)(43 143 109 118 86)(44 119 144 87 110)(45 88 120 111 137)(46 112 81 138 113)(47 139 105 114 82)(48 115 140 83 106)
(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)
G:=sub<Sym(160)| (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(41,127)(42,128)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(129,140)(130,141)(131,142)(132,143)(133,144)(134,137)(135,138)(136,139), (1,48)(2,29)(3,42)(4,31)(5,44)(6,25)(7,46)(8,27)(9,151)(10,88)(11,145)(12,82)(13,147)(14,84)(15,149)(16,86)(17,92)(18,105)(19,94)(20,107)(21,96)(22,109)(23,90)(24,111)(26,128)(28,122)(30,124)(32,126)(33,135)(34,58)(35,129)(36,60)(37,131)(38,62)(39,133)(40,64)(41,123)(43,125)(45,127)(47,121)(49,85)(50,150)(51,87)(52,152)(53,81)(54,146)(55,83)(56,148)(57,142)(59,144)(61,138)(63,140)(65,97)(66,160)(67,99)(68,154)(69,101)(70,156)(71,103)(72,158)(73,112)(74,93)(75,106)(76,95)(77,108)(78,89)(79,110)(80,91)(98,116)(100,118)(102,120)(104,114)(113,157)(115,159)(117,153)(119,155)(130,137)(132,139)(134,141)(136,143), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,142)(34,143)(35,144)(36,137)(37,138)(38,139)(39,140)(40,141)(57,135)(58,136)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,109)(90,110)(91,111)(92,112)(93,105)(94,106)(95,107)(96,108)(97,159)(98,160)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158), (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), (1,159,63,55,75)(2,56,160,76,64)(3,77,49,57,153)(4,58,78,154,50)(5,155,59,51,79)(6,52,156,80,60)(7,73,53,61,157)(8,62,74,158,54)(9,23,126,101,129)(10,102,24,130,127)(11,131,103,128,17)(12,121,132,18,104)(13,19,122,97,133)(14,98,20,134,123)(15,135,99,124,21)(16,125,136,22,100)(25,152,70,91,36)(26,92,145,37,71)(27,38,93,72,146)(28,65,39,147,94)(29,148,66,95,40)(30,96,149,33,67)(31,34,89,68,150)(32,69,35,151,90)(41,84,116,107,141)(42,108,85,142,117)(43,143,109,118,86)(44,119,144,87,110)(45,88,120,111,137)(46,112,81,138,113)(47,139,105,114,82)(48,115,140,83,106), (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)>;
G:=Group( (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,81)(16,82)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(41,127)(42,128)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(129,140)(130,141)(131,142)(132,143)(133,144)(134,137)(135,138)(136,139), (1,48)(2,29)(3,42)(4,31)(5,44)(6,25)(7,46)(8,27)(9,151)(10,88)(11,145)(12,82)(13,147)(14,84)(15,149)(16,86)(17,92)(18,105)(19,94)(20,107)(21,96)(22,109)(23,90)(24,111)(26,128)(28,122)(30,124)(32,126)(33,135)(34,58)(35,129)(36,60)(37,131)(38,62)(39,133)(40,64)(41,123)(43,125)(45,127)(47,121)(49,85)(50,150)(51,87)(52,152)(53,81)(54,146)(55,83)(56,148)(57,142)(59,144)(61,138)(63,140)(65,97)(66,160)(67,99)(68,154)(69,101)(70,156)(71,103)(72,158)(73,112)(74,93)(75,106)(76,95)(77,108)(78,89)(79,110)(80,91)(98,116)(100,118)(102,120)(104,114)(113,157)(115,159)(117,153)(119,155)(130,137)(132,139)(134,141)(136,143), (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,121)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,142)(34,143)(35,144)(36,137)(37,138)(38,139)(39,140)(40,141)(57,135)(58,136)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,109)(90,110)(91,111)(92,112)(93,105)(94,106)(95,107)(96,108)(97,159)(98,160)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158), (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), (1,159,63,55,75)(2,56,160,76,64)(3,77,49,57,153)(4,58,78,154,50)(5,155,59,51,79)(6,52,156,80,60)(7,73,53,61,157)(8,62,74,158,54)(9,23,126,101,129)(10,102,24,130,127)(11,131,103,128,17)(12,121,132,18,104)(13,19,122,97,133)(14,98,20,134,123)(15,135,99,124,21)(16,125,136,22,100)(25,152,70,91,36)(26,92,145,37,71)(27,38,93,72,146)(28,65,39,147,94)(29,148,66,95,40)(30,96,149,33,67)(31,34,89,68,150)(32,69,35,151,90)(41,84,116,107,141)(42,108,85,142,117)(43,143,109,118,86)(44,119,144,87,110)(45,88,120,111,137)(46,112,81,138,113)(47,139,105,114,82)(48,115,140,83,106), (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) );
G=PermutationGroup([[(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,81),(16,82),(17,108),(18,109),(19,110),(20,111),(21,112),(22,105),(23,106),(24,107),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60),(41,127),(42,128),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,153),(72,154),(73,96),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(97,119),(98,120),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(129,140),(130,141),(131,142),(132,143),(133,144),(134,137),(135,138),(136,139)], [(1,48),(2,29),(3,42),(4,31),(5,44),(6,25),(7,46),(8,27),(9,151),(10,88),(11,145),(12,82),(13,147),(14,84),(15,149),(16,86),(17,92),(18,105),(19,94),(20,107),(21,96),(22,109),(23,90),(24,111),(26,128),(28,122),(30,124),(32,126),(33,135),(34,58),(35,129),(36,60),(37,131),(38,62),(39,133),(40,64),(41,123),(43,125),(45,127),(47,121),(49,85),(50,150),(51,87),(52,152),(53,81),(54,146),(55,83),(56,148),(57,142),(59,144),(61,138),(63,140),(65,97),(66,160),(67,99),(68,154),(69,101),(70,156),(71,103),(72,158),(73,112),(74,93),(75,106),(76,95),(77,108),(78,89),(79,110),(80,91),(98,116),(100,118),(102,120),(104,114),(113,157),(115,159),(117,153),(119,155),(130,137),(132,139),(134,141),(136,143)], [(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,121),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,49),(16,50),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(33,142),(34,143),(35,144),(36,137),(37,138),(38,139),(39,140),(40,141),(57,135),(58,136),(59,129),(60,130),(61,131),(62,132),(63,133),(64,134),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,113),(72,114),(81,145),(82,146),(83,147),(84,148),(85,149),(86,150),(87,151),(88,152),(89,109),(90,110),(91,111),(92,112),(93,105),(94,106),(95,107),(96,108),(97,159),(98,160),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158)], [(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)], [(1,159,63,55,75),(2,56,160,76,64),(3,77,49,57,153),(4,58,78,154,50),(5,155,59,51,79),(6,52,156,80,60),(7,73,53,61,157),(8,62,74,158,54),(9,23,126,101,129),(10,102,24,130,127),(11,131,103,128,17),(12,121,132,18,104),(13,19,122,97,133),(14,98,20,134,123),(15,135,99,124,21),(16,125,136,22,100),(25,152,70,91,36),(26,92,145,37,71),(27,38,93,72,146),(28,65,39,147,94),(29,148,66,95,40),(30,96,149,33,67),(31,34,89,68,150),(32,69,35,151,90),(41,84,116,107,141),(42,108,85,142,117),(43,143,109,118,86),(44,119,144,87,110),(45,88,120,111,137),(46,112,81,138,113),(47,139,105,114,82),(48,115,140,83,106)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8P | 10A | ··· | 10O |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | F5 | C5⋊C8 | C2×F5 | C22.F5 | C22⋊F5 |
| kernel | C2×C23.2F5 | C23.2F5 | C22×C5⋊C8 | C23×Dic5 | C22×Dic5 | C23×C10 | C22×C10 | C2×Dic5 | C2×C10 | C24 | C23 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 16 | 4 | 4 | 1 | 4 | 3 | 4 | 4 |
Matrix representation of C2×C23.2F5 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 39 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,37,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18],[0,16,0,0,0,0,0,0,39,0,0,0,0,0,0,0,0,0,9,36,0,0,0,0,0,0,16,32,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C2×C23.2F5 in GAP, Magma, Sage, TeX
C_2\times C_2^3._2F_5
% in TeX
G:=Group("C2xC2^3.2F5"); // GroupNames label
G:=SmallGroup(320,1135);
// by ID
G=gap.SmallGroup(320,1135);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=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^-1=e^3>;
// generators/relations