metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.49D10, C23.45D20, C10.74(C4×D4), (C2×Dic5)⋊17D4, C23.26(C4×D5), C10.32C22≀C2, (C23×Dic5)⋊1C2, C22.44(C2×D20), (C22×C10).68D4, C22.103(D4×D5), (C22×C4).33D10, C2.5(C22⋊D20), C10.86(C4⋊D4), C5⋊4(C23.23D4), C23.36(C5⋊D4), C22⋊1(D10⋊C4), C2.4(Dic5⋊D4), (C22×C20).26C22, (C23×C10).41C22, (C23×D5).14C22, C23.285(C22×D5), C10.10C42⋊16C2, C2.29(Dic5⋊4D4), C22.50(D4⋊2D5), (C22×C10).332C23, C2.5(C22.D20), C10.33(C22.D4), (C22×Dic5).209C22, (C2×C5⋊D4)⋊11C4, (C2×C22⋊C4)⋊5D5, (C10×C22⋊C4)⋊3C2, (C2×Dic5)⋊8(C2×C4), (C22×D5)⋊6(C2×C4), (C2×D10⋊C4)⋊6C2, C22.129(C2×C4×D5), (C2×C10)⋊5(C22⋊C4), (C2×C10).324(C2×D4), C10.78(C2×C22⋊C4), (C22×C5⋊D4).4C2, C22.53(C2×C5⋊D4), C2.10(C2×D10⋊C4), (C2×C10).147(C4○D4), (C2×C10).212(C22×C4), (C22×C10).123(C2×C4), SmallGroup(320,585)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.49D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, af=fa, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 1118 in 286 conjugacy classes, 83 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C5, C2×C4 [×26], D4 [×8], C23, C23 [×6], C23 [×12], D5 [×2], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, Dic5 [×6], C20 [×2], D10 [×10], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×14], C5⋊D4 [×8], C2×C20 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.23D4, D10⋊C4 [×4], C5×C22⋊C4 [×2], C22×Dic5, C22×Dic5 [×2], C22×Dic5 [×6], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, C10.10C42 [×2], C2×D10⋊C4 [×2], C10×C22⋊C4, C23×Dic5, C22×C5⋊D4, C24.49D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.23D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4, Dic5⋊4D4 [×2], C22⋊D20, C22.D20, C2×D10⋊C4, Dic5⋊D4 [×2], C24.49D10
►On 160 points
Generators in S160
(1 121)(2 43)(3 123)(4 45)(5 125)(6 47)(7 127)(8 49)(9 129)(10 51)(11 131)(12 53)(13 133)(14 55)(15 135)(16 57)(17 137)(18 59)(19 139)(20 41)(21 67)(22 148)(23 69)(24 150)(25 71)(26 152)(27 73)(28 154)(29 75)(30 156)(31 77)(32 158)(33 79)(34 160)(35 61)(36 142)(37 63)(38 144)(39 65)(40 146)(42 113)(44 115)(46 117)(48 119)(50 101)(52 103)(54 105)(56 107)(58 109)(60 111)(62 92)(64 94)(66 96)(68 98)(70 100)(72 82)(74 84)(76 86)(78 88)(80 90)(81 151)(83 153)(85 155)(87 157)(89 159)(91 141)(93 143)(95 145)(97 147)(99 149)(102 130)(104 132)(106 134)(108 136)(110 138)(112 140)(114 122)(116 124)(118 126)(120 128)
(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 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 97)(22 98)(23 99)(24 100)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 140)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(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 10 89 98)(2 97 90 9)(3 8 91 96)(4 95 92 7)(5 6 93 94)(11 20 99 88)(12 87 100 19)(13 18 81 86)(14 85 82 17)(15 16 83 84)(21 34 101 114)(22 113 102 33)(23 32 103 112)(24 111 104 31)(25 30 105 110)(26 109 106 29)(27 28 107 108)(35 40 115 120)(36 119 116 39)(37 38 117 118)(41 149 78 131)(42 130 79 148)(43 147 80 129)(44 128 61 146)(45 145 62 127)(46 126 63 144)(47 143 64 125)(48 124 65 142)(49 141 66 123)(50 122 67 160)(51 159 68 121)(52 140 69 158)(53 157 70 139)(54 138 71 156)(55 155 72 137)(56 136 73 154)(57 153 74 135)(58 134 75 152)(59 151 76 133)(60 132 77 150)
G:=sub<Sym(160)| (1,121)(2,43)(3,123)(4,45)(5,125)(6,47)(7,127)(8,49)(9,129)(10,51)(11,131)(12,53)(13,133)(14,55)(15,135)(16,57)(17,137)(18,59)(19,139)(20,41)(21,67)(22,148)(23,69)(24,150)(25,71)(26,152)(27,73)(28,154)(29,75)(30,156)(31,77)(32,158)(33,79)(34,160)(35,61)(36,142)(37,63)(38,144)(39,65)(40,146)(42,113)(44,115)(46,117)(48,119)(50,101)(52,103)(54,105)(56,107)(58,109)(60,111)(62,92)(64,94)(66,96)(68,98)(70,100)(72,82)(74,84)(76,86)(78,88)(80,90)(81,151)(83,153)(85,155)(87,157)(89,159)(91,141)(93,143)(95,145)(97,147)(99,149)(102,130)(104,132)(106,134)(108,136)(110,138)(112,140)(114,122)(116,124)(118,126)(120,128), (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), (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,97)(22,98)(23,99)(24,100)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,140)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (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), (1,10,89,98)(2,97,90,9)(3,8,91,96)(4,95,92,7)(5,6,93,94)(11,20,99,88)(12,87,100,19)(13,18,81,86)(14,85,82,17)(15,16,83,84)(21,34,101,114)(22,113,102,33)(23,32,103,112)(24,111,104,31)(25,30,105,110)(26,109,106,29)(27,28,107,108)(35,40,115,120)(36,119,116,39)(37,38,117,118)(41,149,78,131)(42,130,79,148)(43,147,80,129)(44,128,61,146)(45,145,62,127)(46,126,63,144)(47,143,64,125)(48,124,65,142)(49,141,66,123)(50,122,67,160)(51,159,68,121)(52,140,69,158)(53,157,70,139)(54,138,71,156)(55,155,72,137)(56,136,73,154)(57,153,74,135)(58,134,75,152)(59,151,76,133)(60,132,77,150)>;
G:=Group( (1,121)(2,43)(3,123)(4,45)(5,125)(6,47)(7,127)(8,49)(9,129)(10,51)(11,131)(12,53)(13,133)(14,55)(15,135)(16,57)(17,137)(18,59)(19,139)(20,41)(21,67)(22,148)(23,69)(24,150)(25,71)(26,152)(27,73)(28,154)(29,75)(30,156)(31,77)(32,158)(33,79)(34,160)(35,61)(36,142)(37,63)(38,144)(39,65)(40,146)(42,113)(44,115)(46,117)(48,119)(50,101)(52,103)(54,105)(56,107)(58,109)(60,111)(62,92)(64,94)(66,96)(68,98)(70,100)(72,82)(74,84)(76,86)(78,88)(80,90)(81,151)(83,153)(85,155)(87,157)(89,159)(91,141)(93,143)(95,145)(97,147)(99,149)(102,130)(104,132)(106,134)(108,136)(110,138)(112,140)(114,122)(116,124)(118,126)(120,128), (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), (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,97)(22,98)(23,99)(24,100)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,140)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (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), (1,10,89,98)(2,97,90,9)(3,8,91,96)(4,95,92,7)(5,6,93,94)(11,20,99,88)(12,87,100,19)(13,18,81,86)(14,85,82,17)(15,16,83,84)(21,34,101,114)(22,113,102,33)(23,32,103,112)(24,111,104,31)(25,30,105,110)(26,109,106,29)(27,28,107,108)(35,40,115,120)(36,119,116,39)(37,38,117,118)(41,149,78,131)(42,130,79,148)(43,147,80,129)(44,128,61,146)(45,145,62,127)(46,126,63,144)(47,143,64,125)(48,124,65,142)(49,141,66,123)(50,122,67,160)(51,159,68,121)(52,140,69,158)(53,157,70,139)(54,138,71,156)(55,155,72,137)(56,136,73,154)(57,153,74,135)(58,134,75,152)(59,151,76,133)(60,132,77,150) );
G=PermutationGroup([(1,121),(2,43),(3,123),(4,45),(5,125),(6,47),(7,127),(8,49),(9,129),(10,51),(11,131),(12,53),(13,133),(14,55),(15,135),(16,57),(17,137),(18,59),(19,139),(20,41),(21,67),(22,148),(23,69),(24,150),(25,71),(26,152),(27,73),(28,154),(29,75),(30,156),(31,77),(32,158),(33,79),(34,160),(35,61),(36,142),(37,63),(38,144),(39,65),(40,146),(42,113),(44,115),(46,117),(48,119),(50,101),(52,103),(54,105),(56,107),(58,109),(60,111),(62,92),(64,94),(66,96),(68,98),(70,100),(72,82),(74,84),(76,86),(78,88),(80,90),(81,151),(83,153),(85,155),(87,157),(89,159),(91,141),(93,143),(95,145),(97,147),(99,149),(102,130),(104,132),(106,134),(108,136),(110,138),(112,140),(114,122),(116,124),(118,126),(120,128)], [(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)], [(1,99),(2,100),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,97),(22,98),(23,99),(24,100),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,140),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(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)], [(1,10,89,98),(2,97,90,9),(3,8,91,96),(4,95,92,7),(5,6,93,94),(11,20,99,88),(12,87,100,19),(13,18,81,86),(14,85,82,17),(15,16,83,84),(21,34,101,114),(22,113,102,33),(23,32,103,112),(24,111,104,31),(25,30,105,110),(26,109,106,29),(27,28,107,108),(35,40,115,120),(36,119,116,39),(37,38,117,118),(41,149,78,131),(42,130,79,148),(43,147,80,129),(44,128,61,146),(45,145,62,127),(46,126,63,144),(47,143,64,125),(48,124,65,142),(49,141,66,123),(50,122,67,160),(51,159,68,121),(52,140,69,158),(53,157,70,139),(54,138,71,156),(55,155,72,137),(56,136,73,154),(57,153,74,135),(58,134,75,152),(59,151,76,133),(60,132,77,150)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C24.49D10 | C10.10C42 | C2×D10⋊C4 | C10×C22⋊C4 | C23×Dic5 | C22×C5⋊D4 | C2×C5⋊D4 | C2×Dic5 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 2 | 4 | 4 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C24.49D10 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 9 |
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 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 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 14 | 27 | 0 | 0 |
| 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 2 | 9 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 27 | 14 | 0 | 0 |
| 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,9,40],[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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[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],[32,0,0,0,0,0,14,14,0,0,0,27,30,0,0,0,0,0,32,2,0,0,0,0,9],[9,0,0,0,0,0,27,30,0,0,0,14,14,0,0,0,0,0,32,0,0,0,0,0,32] >;
C24.49D10 in GAP, Magma, Sage, TeX
C_2^4._{49}D_{10} % in TeX
G:=Group("C2^4.49D10"); // GroupNames label
G:=SmallGroup(320,585);
// by ID
G=gap.SmallGroup(320,585);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,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