direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.8D10, C20.34C24, D20.30C23, Dic10.29C23, C4○D4⋊15D10, C10⋊5(C4○D8), D4⋊D5⋊23C22, (C2×C20).502D4, C20.263(C2×D4), Q8⋊D5⋊21C22, C4.34(C23×D5), (C2×D4).232D10, C4○D20⋊20C22, C5⋊2C8.30C23, D4.D5⋊20C22, (C2×Q8).190D10, (C5×D4).22C23, C5⋊Q16⋊20C22, D4.22(C22×D5), Q8.22(C22×D5), (C5×Q8).22C23, (C2×C20).556C23, C10.159(C22×D4), (C22×C4).387D10, (C22×C10).123D4, C23.45(C5⋊D4), (D4×C10).272C22, (C2×D20).288C22, (Q8×C10).237C22, (C22×C20).291C22, (C2×Dic10).317C22, C5⋊6(C2×C4○D8), (C2×C4○D4)⋊3D5, (C2×D4⋊D5)⋊33C2, (C10×C4○D4)⋊3C2, (C2×Q8⋊D5)⋊33C2, C4.30(C2×C5⋊D4), (C2×C4○D20)⋊30C2, (C2×D4.D5)⋊33C2, (C2×C5⋊Q16)⋊33C2, (C2×C10).76(C2×D4), (C22×C5⋊2C8)⋊15C2, (C2×C5⋊2C8)⋊42C22, (C5×C4○D4)⋊17C22, C2.32(C22×C5⋊D4), (C2×C4).158(C5⋊D4), (C2×C4).636(C22×D5), C22.119(C2×C5⋊D4), SmallGroup(320,1493)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 830 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×6], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, C5⋊2C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, C22×C10, C2×C4○D8, C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×4], D4⋊D5 [×4], D4.D5 [×4], Q8⋊D5 [×4], C5⋊Q16 [×4], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C22×C5⋊2C8, C2×D4⋊D5, C2×D4.D5, C2×Q8⋊D5, C2×C5⋊Q16, D4.8D10 [×8], C2×C4○D20, C10×C4○D4, C2×D4.8D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C4○D8, C2×C5⋊D4 [×6], C23×D5, D4.8D10 [×2], C22×C5⋊D4, C2×D4.8D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d9 >
►On 160 points
Generators in S160
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 141)(39 142)(40 143)(41 80)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 121)(99 122)(100 123)
(1 156 11 146)(2 157 12 147)(3 158 13 148)(4 159 14 149)(5 160 15 150)(6 141 16 151)(7 142 17 152)(8 143 18 153)(9 144 19 154)(10 145 20 155)(21 108 31 118)(22 109 32 119)(23 110 33 120)(24 111 34 101)(25 112 35 102)(26 113 36 103)(27 114 37 104)(28 115 38 105)(29 116 39 106)(30 117 40 107)(41 129 51 139)(42 130 52 140)(43 131 53 121)(44 132 54 122)(45 133 55 123)(46 134 56 124)(47 135 57 125)(48 136 58 126)(49 137 59 127)(50 138 60 128)(61 87 71 97)(62 88 72 98)(63 89 73 99)(64 90 74 100)(65 91 75 81)(66 92 76 82)(67 93 77 83)(68 94 78 84)(69 95 79 85)(70 96 80 86)
(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 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 101)(78 102)(79 103)(80 104)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 141)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 140 51 130)(42 129 52 139)(43 138 53 128)(44 127 54 137)(45 136 55 126)(46 125 56 135)(47 134 57 124)(48 123 58 133)(49 132 59 122)(50 121 60 131)(61 86 71 96)(62 95 72 85)(63 84 73 94)(64 93 74 83)(65 82 75 92)(66 91 76 81)(67 100 77 90)(68 89 78 99)(69 98 79 88)(70 87 80 97)(101 118 111 108)(102 107 112 117)(103 116 113 106)(104 105 114 115)(109 110 119 120)(141 160 151 150)(142 149 152 159)(143 158 153 148)(144 147 154 157)(145 156 155 146)
G:=sub<Sym(160)| (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,141)(39,142)(40,143)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (1,156,11,146)(2,157,12,147)(3,158,13,148)(4,159,14,149)(5,160,15,150)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(21,108,31,118)(22,109,32,119)(23,110,33,120)(24,111,34,101)(25,112,35,102)(26,113,36,103)(27,114,37,104)(28,115,38,105)(29,116,39,106)(30,117,40,107)(41,129,51,139)(42,130,52,140)(43,131,53,121)(44,132,54,122)(45,133,55,123)(46,134,56,124)(47,135,57,125)(48,136,58,126)(49,137,59,127)(50,138,60,128)(61,87,71,97)(62,88,72,98)(63,89,73,99)(64,90,74,100)(65,91,75,81)(66,92,76,82)(67,93,77,83)(68,94,78,84)(69,95,79,85)(70,96,80,86), (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)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,141)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,140,51,130)(42,129,52,139)(43,138,53,128)(44,127,54,137)(45,136,55,126)(46,125,56,135)(47,134,57,124)(48,123,58,133)(49,132,59,122)(50,121,60,131)(61,86,71,96)(62,95,72,85)(63,84,73,94)(64,93,74,83)(65,82,75,92)(66,91,76,81)(67,100,77,90)(68,89,78,99)(69,98,79,88)(70,87,80,97)(101,118,111,108)(102,107,112,117)(103,116,113,106)(104,105,114,115)(109,110,119,120)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146)>;
G:=Group( (1,110)(2,111)(3,112)(4,113)(5,114)(6,115)(7,116)(8,117)(9,118)(10,119)(11,120)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,141)(39,142)(40,143)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (1,156,11,146)(2,157,12,147)(3,158,13,148)(4,159,14,149)(5,160,15,150)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(21,108,31,118)(22,109,32,119)(23,110,33,120)(24,111,34,101)(25,112,35,102)(26,113,36,103)(27,114,37,104)(28,115,38,105)(29,116,39,106)(30,117,40,107)(41,129,51,139)(42,130,52,140)(43,131,53,121)(44,132,54,122)(45,133,55,123)(46,134,56,124)(47,135,57,125)(48,136,58,126)(49,137,59,127)(50,138,60,128)(61,87,71,97)(62,88,72,98)(63,89,73,99)(64,90,74,100)(65,91,75,81)(66,92,76,82)(67,93,77,83)(68,94,78,84)(69,95,79,85)(70,96,80,86), (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)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,141)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,140,51,130)(42,129,52,139)(43,138,53,128)(44,127,54,137)(45,136,55,126)(46,125,56,135)(47,134,57,124)(48,123,58,133)(49,132,59,122)(50,121,60,131)(61,86,71,96)(62,95,72,85)(63,84,73,94)(64,93,74,83)(65,82,75,92)(66,91,76,81)(67,100,77,90)(68,89,78,99)(69,98,79,88)(70,87,80,97)(101,118,111,108)(102,107,112,117)(103,116,113,106)(104,105,114,115)(109,110,119,120)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146) );
G=PermutationGroup([(1,110),(2,111),(3,112),(4,113),(5,114),(6,115),(7,116),(8,117),(9,118),(10,119),(11,120),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,144),(22,145),(23,146),(24,147),(25,148),(26,149),(27,150),(28,151),(29,152),(30,153),(31,154),(32,155),(33,156),(34,157),(35,158),(36,159),(37,160),(38,141),(39,142),(40,143),(41,80),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,121),(99,122),(100,123)], [(1,156,11,146),(2,157,12,147),(3,158,13,148),(4,159,14,149),(5,160,15,150),(6,141,16,151),(7,142,17,152),(8,143,18,153),(9,144,19,154),(10,145,20,155),(21,108,31,118),(22,109,32,119),(23,110,33,120),(24,111,34,101),(25,112,35,102),(26,113,36,103),(27,114,37,104),(28,115,38,105),(29,116,39,106),(30,117,40,107),(41,129,51,139),(42,130,52,140),(43,131,53,121),(44,132,54,122),(45,133,55,123),(46,134,56,124),(47,135,57,125),(48,136,58,126),(49,137,59,127),(50,138,60,128),(61,87,71,97),(62,88,72,98),(63,89,73,99),(64,90,74,100),(65,91,75,81),(66,92,76,82),(67,93,77,83),(68,94,78,84),(69,95,79,85),(70,96,80,86)], [(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),(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,101),(78,102),(79,103),(80,104),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,141),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,151)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,24,31,34),(22,33,32,23),(25,40,35,30),(26,29,36,39),(27,38,37,28),(41,140,51,130),(42,129,52,139),(43,138,53,128),(44,127,54,137),(45,136,55,126),(46,125,56,135),(47,134,57,124),(48,123,58,133),(49,132,59,122),(50,121,60,131),(61,86,71,96),(62,95,72,85),(63,84,73,94),(64,93,74,83),(65,82,75,92),(66,91,76,81),(67,100,77,90),(68,89,78,99),(69,98,79,88),(70,87,80,97),(101,118,111,108),(102,107,112,117),(103,116,113,106),(104,105,114,115),(109,110,119,120),(141,160,151,150),(142,149,152,159),(143,158,153,148),(144,147,154,157),(145,156,155,146)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 0 | 0 | 12 | 24 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1,0,0,0,0,0,0,17,12,0,0,0,0,17,24],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,0,32] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | C2×D4.8D10 | C22×C5⋊2C8 | C2×D4⋊D5 | C2×D4.D5 | C2×Q8⋊D5 | C2×C5⋊Q16 | D4.8D10 | C2×C4○D20 | C10×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 12 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4._8D_{10} % in TeX
G:=Group("C2xD4.8D10"); // GroupNames label
G:=SmallGroup(320,1493);
// by ID
G=gap.SmallGroup(320,1493);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^9>;
// generators/relations
×
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ℤ
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