metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.10D10, C23.14D20, (C2×C20).51D4, C2.6(C20⋊7D4), (C22×C4).91D10, (C22×C10).65D4, C10.58(C4⋊D4), C22.125(C2×D20), C5⋊4(C23.11D4), C2.6(C20.17D4), C10.38(C4.4D4), C22.98(C4○D20), (C22×C20).60C22, (C23×C10).36C22, C23.370(C22×D5), C10.10C42⋊15C2, C10.16(C42⋊2C2), C22.96(D4⋊2D5), (C22×C10).328C23, C2.14(C23.D10), C2.8(C23.18D10), C10.73(C22.D4), C2.16(C22.D20), (C22×Dic5).42C22, (C2×C4⋊Dic5)⋊12C2, (C2×C10).432(C2×D4), (C2×C4).30(C5⋊D4), (C2×C22⋊C4).15D5, (C10×C22⋊C4).16C2, C22.126(C2×C5⋊D4), (C2×C23.D5).15C2, (C2×C10).144(C4○D4), SmallGroup(320,580)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.10D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 566 in 170 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×17], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×4], C20 [×3], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic5 [×12], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C23.11D4, C4⋊Dic5 [×2], C23.D5 [×4], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42, C10.10C42 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C24.10D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4 [×5], D10 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.11D4, C2×D20, C4○D20, D4⋊2D5 [×4], C2×C5⋊D4, C23.D10 [×2], C22.D20 [×2], C20⋊7D4, C23.18D10, C20.17D4, C24.10D10
►On 160 points
Generators in S160
(1 11)(2 132)(3 13)(4 134)(5 15)(6 136)(7 17)(8 138)(9 19)(10 140)(12 122)(14 124)(16 126)(18 128)(20 130)(21 159)(22 93)(23 141)(24 95)(25 143)(26 97)(27 145)(28 99)(29 147)(30 81)(31 149)(32 83)(33 151)(34 85)(35 153)(36 87)(37 155)(38 89)(39 157)(40 91)(41 51)(42 105)(43 53)(44 107)(45 55)(46 109)(47 57)(48 111)(49 59)(50 113)(52 115)(54 117)(56 119)(58 101)(60 103)(61 148)(62 82)(63 150)(64 84)(65 152)(66 86)(67 154)(68 88)(69 156)(70 90)(71 158)(72 92)(73 160)(74 94)(75 142)(76 96)(77 144)(78 98)(79 146)(80 100)(102 112)(104 114)(106 116)(108 118)(110 120)(121 131)(123 133)(125 135)(127 137)(129 139)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 121)
(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)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)
(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 73 121 22)(2 72 122 21)(3 71 123 40)(4 70 124 39)(5 69 125 38)(6 68 126 37)(7 67 127 36)(8 66 128 35)(9 65 129 34)(10 64 130 33)(11 63 131 32)(12 62 132 31)(13 61 133 30)(14 80 134 29)(15 79 135 28)(16 78 136 27)(17 77 137 26)(18 76 138 25)(19 75 139 24)(20 74 140 23)(41 89 114 156)(42 88 115 155)(43 87 116 154)(44 86 117 153)(45 85 118 152)(46 84 119 151)(47 83 120 150)(48 82 101 149)(49 81 102 148)(50 100 103 147)(51 99 104 146)(52 98 105 145)(53 97 106 144)(54 96 107 143)(55 95 108 142)(56 94 109 141)(57 93 110 160)(58 92 111 159)(59 91 112 158)(60 90 113 157)
G:=sub<Sym(160)| (1,11)(2,132)(3,13)(4,134)(5,15)(6,136)(7,17)(8,138)(9,19)(10,140)(12,122)(14,124)(16,126)(18,128)(20,130)(21,159)(22,93)(23,141)(24,95)(25,143)(26,97)(27,145)(28,99)(29,147)(30,81)(31,149)(32,83)(33,151)(34,85)(35,153)(36,87)(37,155)(38,89)(39,157)(40,91)(41,51)(42,105)(43,53)(44,107)(45,55)(46,109)(47,57)(48,111)(49,59)(50,113)(52,115)(54,117)(56,119)(58,101)(60,103)(61,148)(62,82)(63,150)(64,84)(65,152)(66,86)(67,154)(68,88)(69,156)(70,90)(71,158)(72,92)(73,160)(74,94)(75,142)(76,96)(77,144)(78,98)(79,146)(80,100)(102,112)(104,114)(106,116)(108,118)(110,120)(121,131)(123,133)(125,135)(127,137)(129,139), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,121), (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), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147), (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), (1,73,121,22)(2,72,122,21)(3,71,123,40)(4,70,124,39)(5,69,125,38)(6,68,126,37)(7,67,127,36)(8,66,128,35)(9,65,129,34)(10,64,130,33)(11,63,131,32)(12,62,132,31)(13,61,133,30)(14,80,134,29)(15,79,135,28)(16,78,136,27)(17,77,137,26)(18,76,138,25)(19,75,139,24)(20,74,140,23)(41,89,114,156)(42,88,115,155)(43,87,116,154)(44,86,117,153)(45,85,118,152)(46,84,119,151)(47,83,120,150)(48,82,101,149)(49,81,102,148)(50,100,103,147)(51,99,104,146)(52,98,105,145)(53,97,106,144)(54,96,107,143)(55,95,108,142)(56,94,109,141)(57,93,110,160)(58,92,111,159)(59,91,112,158)(60,90,113,157)>;
G:=Group( (1,11)(2,132)(3,13)(4,134)(5,15)(6,136)(7,17)(8,138)(9,19)(10,140)(12,122)(14,124)(16,126)(18,128)(20,130)(21,159)(22,93)(23,141)(24,95)(25,143)(26,97)(27,145)(28,99)(29,147)(30,81)(31,149)(32,83)(33,151)(34,85)(35,153)(36,87)(37,155)(38,89)(39,157)(40,91)(41,51)(42,105)(43,53)(44,107)(45,55)(46,109)(47,57)(48,111)(49,59)(50,113)(52,115)(54,117)(56,119)(58,101)(60,103)(61,148)(62,82)(63,150)(64,84)(65,152)(66,86)(67,154)(68,88)(69,156)(70,90)(71,158)(72,92)(73,160)(74,94)(75,142)(76,96)(77,144)(78,98)(79,146)(80,100)(102,112)(104,114)(106,116)(108,118)(110,120)(121,131)(123,133)(125,135)(127,137)(129,139), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,121), (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), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147), (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), (1,73,121,22)(2,72,122,21)(3,71,123,40)(4,70,124,39)(5,69,125,38)(6,68,126,37)(7,67,127,36)(8,66,128,35)(9,65,129,34)(10,64,130,33)(11,63,131,32)(12,62,132,31)(13,61,133,30)(14,80,134,29)(15,79,135,28)(16,78,136,27)(17,77,137,26)(18,76,138,25)(19,75,139,24)(20,74,140,23)(41,89,114,156)(42,88,115,155)(43,87,116,154)(44,86,117,153)(45,85,118,152)(46,84,119,151)(47,83,120,150)(48,82,101,149)(49,81,102,148)(50,100,103,147)(51,99,104,146)(52,98,105,145)(53,97,106,144)(54,96,107,143)(55,95,108,142)(56,94,109,141)(57,93,110,160)(58,92,111,159)(59,91,112,158)(60,90,113,157) );
G=PermutationGroup([(1,11),(2,132),(3,13),(4,134),(5,15),(6,136),(7,17),(8,138),(9,19),(10,140),(12,122),(14,124),(16,126),(18,128),(20,130),(21,159),(22,93),(23,141),(24,95),(25,143),(26,97),(27,145),(28,99),(29,147),(30,81),(31,149),(32,83),(33,151),(34,85),(35,153),(36,87),(37,155),(38,89),(39,157),(40,91),(41,51),(42,105),(43,53),(44,107),(45,55),(46,109),(47,57),(48,111),(49,59),(50,113),(52,115),(54,117),(56,119),(58,101),(60,103),(61,148),(62,82),(63,150),(64,84),(65,152),(66,86),(67,154),(68,88),(69,156),(70,90),(71,158),(72,92),(73,160),(74,94),(75,142),(76,96),(77,144),(78,98),(79,146),(80,100),(102,112),(104,114),(106,116),(108,118),(110,120),(121,131),(123,133),(125,135),(127,137),(129,139)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,149),(22,150),(23,151),(24,152),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,141),(34,142),(35,143),(36,144),(37,145),(38,146),(39,147),(40,148),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,121)], [(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)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113),(81,148),(82,149),(83,150),(84,151),(85,152),(86,153),(87,154),(88,155),(89,156),(90,157),(91,158),(92,159),(93,160),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147)], [(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)], [(1,73,121,22),(2,72,122,21),(3,71,123,40),(4,70,124,39),(5,69,125,38),(6,68,126,37),(7,67,127,36),(8,66,128,35),(9,65,129,34),(10,64,130,33),(11,63,131,32),(12,62,132,31),(13,61,133,30),(14,80,134,29),(15,79,135,28),(16,78,136,27),(17,77,137,26),(18,76,138,25),(19,75,139,24),(20,74,140,23),(41,89,114,156),(42,88,115,155),(43,87,116,154),(44,86,117,153),(45,85,118,152),(46,84,119,151),(47,83,120,150),(48,82,101,149),(49,81,102,148),(50,100,103,147),(51,99,104,146),(52,98,105,145),(53,97,106,144),(54,96,107,143),(55,95,108,142),(56,94,109,141),(57,93,110,160),(58,92,111,159),(59,91,112,158),(60,90,113,157)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | D4⋊2D5 |
| kernel | C24.10D10 | C10.10C42 | C2×C4⋊Dic5 | C2×C23.D5 | C10×C22⋊C4 | C2×C20 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 2 | 10 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C24.10D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 1 | 0 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 16 | 0 | 0 |
| 0 | 0 | 36 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 6 | 28 | 0 | 0 | 0 | 0 |
| 9 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 27 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[10,38,0,0,0,0,0,37,0,0,0,0,0,0,14,36,0,0,0,0,16,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[6,9,0,0,0,0,28,35,0,0,0,0,0,0,35,27,0,0,0,0,23,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
C24.10D10 in GAP, Magma, Sage, TeX
C_2^4._{10}D_{10} % in TeX
G:=Group("C2^4.10D10"); // GroupNames label
G:=SmallGroup(320,580);
// by ID
G=gap.SmallGroup(320,580);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,387,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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*e^9>;
// generators/relations