direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.D10, C24.22D10, (C2×C10).26C24, C4⋊Dic5⋊50C22, C22⋊C4.85D10, C10⋊2(C42⋊2C2), (C2×C20).126C23, (C4×Dic5)⋊72C22, (C22×C4).168D10, C23.78(C22×D5), C22.68(C23×D5), C22.71(C4○D20), C10.D4⋊47C22, (C23×C10).52C22, C23.D5.84C22, C22.65(D4⋊2D5), (C22×C10).118C23, (C22×C20).350C22, (C2×Dic5).186C23, (C22×Dic5).226C22, C5⋊2(C2×C42⋊2C2), (C2×C4×Dic5)⋊29C2, (C2×C4⋊Dic5)⋊18C2, C10.11(C2×C4○D4), C2.13(C2×C4○D20), C2.8(C2×D4⋊2D5), (C2×C22⋊C4).17D5, (C2×C10.D4)⋊34C2, (C10×C22⋊C4).20C2, (C2×C4).255(C22×D5), (C2×C23.D5).21C2, (C2×C10).100(C4○D4), (C5×C22⋊C4).108C22, SmallGroup(320,1154)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.D10
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, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 686 in 246 conjugacy classes, 111 normal (31 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, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C42⋊2C2, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C23.D10, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C2×C23.D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C42⋊2C2, C2×C4○D4, C22×D5, C2×C42⋊2C2, C4○D20, D4⋊2D5, C23×D5, C23.D10, C2×C4○D20, C2×D4⋊2D5, C2×C23.D10
►On 160 points
Generators in S160
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 81)(37 82)(38 83)(39 84)(40 85)(41 120)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 151)(79 152)(80 153)
(2 75)(4 77)(6 79)(8 61)(10 63)(12 65)(14 67)(16 69)(18 71)(20 73)(21 105)(22 32)(23 107)(24 34)(25 109)(26 36)(27 111)(28 38)(29 113)(30 40)(31 115)(33 117)(35 119)(37 101)(39 103)(41 51)(42 82)(43 53)(44 84)(45 55)(46 86)(47 57)(48 88)(49 59)(50 90)(52 92)(54 94)(56 96)(58 98)(60 100)(81 91)(83 93)(85 95)(87 97)(89 99)(102 112)(104 114)(106 116)(108 118)(110 120)(122 160)(124 142)(126 144)(128 146)(130 148)(132 150)(134 152)(136 154)(138 156)(140 158)
(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 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(121 159)(122 160)(123 141)(124 142)(125 143)(126 144)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)
(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 30 64 114)(2 39 65 103)(3 28 66 112)(4 37 67 101)(5 26 68 110)(6 35 69 119)(7 24 70 108)(8 33 71 117)(9 22 72 106)(10 31 73 115)(11 40 74 104)(12 29 75 113)(13 38 76 102)(14 27 77 111)(15 36 78 120)(16 25 79 109)(17 34 80 118)(18 23 61 107)(19 32 62 116)(20 21 63 105)(41 123 81 151)(42 132 82 160)(43 121 83 149)(44 130 84 158)(45 139 85 147)(46 128 86 156)(47 137 87 145)(48 126 88 154)(49 135 89 143)(50 124 90 152)(51 133 91 141)(52 122 92 150)(53 131 93 159)(54 140 94 148)(55 129 95 157)(56 138 96 146)(57 127 97 155)(58 136 98 144)(59 125 99 153)(60 134 100 142)
G:=sub<Sym(160)| (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(41,120)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153), (2,75)(4,77)(6,79)(8,61)(10,63)(12,65)(14,67)(16,69)(18,71)(20,73)(21,105)(22,32)(23,107)(24,34)(25,109)(26,36)(27,111)(28,38)(29,113)(30,40)(31,115)(33,117)(35,119)(37,101)(39,103)(41,51)(42,82)(43,53)(44,84)(45,55)(46,86)(47,57)(48,88)(49,59)(50,90)(52,92)(54,94)(56,96)(58,98)(60,100)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120)(122,160)(124,142)(126,144)(128,146)(130,148)(132,150)(134,152)(136,154)(138,156)(140,158), (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), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158), (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), (1,30,64,114)(2,39,65,103)(3,28,66,112)(4,37,67,101)(5,26,68,110)(6,35,69,119)(7,24,70,108)(8,33,71,117)(9,22,72,106)(10,31,73,115)(11,40,74,104)(12,29,75,113)(13,38,76,102)(14,27,77,111)(15,36,78,120)(16,25,79,109)(17,34,80,118)(18,23,61,107)(19,32,62,116)(20,21,63,105)(41,123,81,151)(42,132,82,160)(43,121,83,149)(44,130,84,158)(45,139,85,147)(46,128,86,156)(47,137,87,145)(48,126,88,154)(49,135,89,143)(50,124,90,152)(51,133,91,141)(52,122,92,150)(53,131,93,159)(54,140,94,148)(55,129,95,157)(56,138,96,146)(57,127,97,155)(58,136,98,144)(59,125,99,153)(60,134,100,142)>;
G:=Group( (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(41,120)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153), (2,75)(4,77)(6,79)(8,61)(10,63)(12,65)(14,67)(16,69)(18,71)(20,73)(21,105)(22,32)(23,107)(24,34)(25,109)(26,36)(27,111)(28,38)(29,113)(30,40)(31,115)(33,117)(35,119)(37,101)(39,103)(41,51)(42,82)(43,53)(44,84)(45,55)(46,86)(47,57)(48,88)(49,59)(50,90)(52,92)(54,94)(56,96)(58,98)(60,100)(81,91)(83,93)(85,95)(87,97)(89,99)(102,112)(104,114)(106,116)(108,118)(110,120)(122,160)(124,142)(126,144)(128,146)(130,148)(132,150)(134,152)(136,154)(138,156)(140,158), (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), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158), (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), (1,30,64,114)(2,39,65,103)(3,28,66,112)(4,37,67,101)(5,26,68,110)(6,35,69,119)(7,24,70,108)(8,33,71,117)(9,22,72,106)(10,31,73,115)(11,40,74,104)(12,29,75,113)(13,38,76,102)(14,27,77,111)(15,36,78,120)(16,25,79,109)(17,34,80,118)(18,23,61,107)(19,32,62,116)(20,21,63,105)(41,123,81,151)(42,132,82,160)(43,121,83,149)(44,130,84,158)(45,139,85,147)(46,128,86,156)(47,137,87,145)(48,126,88,154)(49,135,89,143)(50,124,90,152)(51,133,91,141)(52,122,92,150)(53,131,93,159)(54,140,94,148)(55,129,95,157)(56,138,96,146)(57,127,97,155)(58,136,98,144)(59,125,99,153)(60,134,100,142) );
G=PermutationGroup([[(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,81),(37,82),(38,83),(39,84),(40,85),(41,120),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,154),(62,155),(63,156),(64,157),(65,158),(66,159),(67,160),(68,141),(69,142),(70,143),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,151),(79,152),(80,153)], [(2,75),(4,77),(6,79),(8,61),(10,63),(12,65),(14,67),(16,69),(18,71),(20,73),(21,105),(22,32),(23,107),(24,34),(25,109),(26,36),(27,111),(28,38),(29,113),(30,40),(31,115),(33,117),(35,119),(37,101),(39,103),(41,51),(42,82),(43,53),(44,84),(45,55),(46,86),(47,57),(48,88),(49,59),(50,90),(52,92),(54,94),(56,96),(58,98),(60,100),(81,91),(83,93),(85,95),(87,97),(89,99),(102,112),(104,114),(106,116),(108,118),(110,120),(122,160),(124,142),(126,144),(128,146),(130,148),(132,150),(134,152),(136,154),(138,156),(140,158)], [(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)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(121,159),(122,160),(123,141),(124,142),(125,143),(126,144),(127,145),(128,146),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,157),(140,158)], [(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)], [(1,30,64,114),(2,39,65,103),(3,28,66,112),(4,37,67,101),(5,26,68,110),(6,35,69,119),(7,24,70,108),(8,33,71,117),(9,22,72,106),(10,31,73,115),(11,40,74,104),(12,29,75,113),(13,38,76,102),(14,27,77,111),(15,36,78,120),(16,25,79,109),(17,34,80,118),(18,23,61,107),(19,32,62,116),(20,21,63,105),(41,123,81,151),(42,132,82,160),(43,121,83,149),(44,130,84,158),(45,139,85,147),(46,128,86,156),(47,137,87,145),(48,126,88,154),(49,135,89,143),(50,124,90,152),(51,133,91,141),(52,122,92,150),(53,131,93,159),(54,140,94,148),(55,129,95,157),(56,138,96,146),(57,127,97,155),(58,136,98,144),(59,125,99,153),(60,134,100,142)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | D4⋊2D5 |
| kernel | C2×C23.D10 | C23.D10 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C2×C23.D5 | C10×C22⋊C4 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 2 | 12 | 8 | 4 | 2 | 16 | 8 |
Matrix representation of C2×C23.D10 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 39 | 40 |
| 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 |
| 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 | 2 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 32 | 32 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,39,0,0,0,0,40],[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],[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],[1,0,0,0,0,0,2,0,0,0,0,0,20,0,0,0,0,0,32,0,0,0,0,32,9],[1,0,0,0,0,0,0,2,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,1,40] >;
C2×C23.D10 in GAP, Magma, Sage, TeX
C_2\times C_2^3.D_{10} % in TeX
G:=Group("C2xC2^3.D10"); // GroupNames label
G:=SmallGroup(320,1154);
// by ID
G=gap.SmallGroup(320,1154);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,297,12550]);
// 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,f*b*f^-1=b*c=c*b,e*b*e^-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