?
G = C10.432+ (1+4) order 320 = 26·5
43rd non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.432+ (1+4), C4⋊D4⋊17D5, C20⋊2D4⋊23C2, C4⋊C4.182D10, (D4×Dic5)⋊22C2, (C2×D4).156D10, C22⋊C4.49D10, C4.Dic10⋊19C2, Dic5⋊4D4⋊11C2, D10.53(C4○D4), C20.203(C4○D4), C4.96(D4⋊2D5), (C2×C10).158C24, (C2×C20).596C23, (C22×C4).225D10, C2.45(D4⋊6D10), C23.18(C22×D5), (D4×C10).124C22, C23.D10⋊19C2, C4⋊Dic5.372C22, (C22×C10).25C23, (C2×Dic5).77C23, C22.179(C23×D5), C23.D5.26C22, C23.21D10⋊26C2, C23.18D10⋊11C2, (C22×C20).243C22, C5⋊7(C22.47C24), (C4×Dic5).104C22, (C22×D5).201C23, D10⋊C4.127C22, C10.D4.139C22, (C22×Dic5).111C22, (D5×C4⋊C4)⋊23C2, (C4×C5⋊D4)⋊19C2, C2.42(D5×C4○D4), (C5×C4⋊D4)⋊20C2, (C2×C4×D5).95C22, C10.155(C2×C4○D4), C2.38(C2×D4⋊2D5), (C2×C4).40(C22×D5), (C5×C4⋊C4).146C22, (C2×C5⋊D4).31C22, (C5×C22⋊C4).15C22, SmallGroup(320,1286)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 790 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×10], C23, C23 [×2], C23, D5 [×2], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], Dic5 [×7], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×9], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×C10, C22×C10 [×2], C22.47C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4, C23.D5, C23.D5 [×6], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.D10 [×2], Dic5⋊4D4 [×2], C4.Dic10, D5×C4⋊C4, C23.21D10, C4×C5⋊D4, D4×Dic5, C23.18D10 [×2], C20⋊2D4, C20⋊2D4 [×2], C5×C4⋊D4, C10.432+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊6D10, D5×C4○D4, C10.432+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, bd=db, ebe-1=a5b, cd=dc, ece-1=a5c, ede-1=b2d >
►On 160 points
Generators in S160
(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 16 25 157)(2 15 26 156)(3 14 27 155)(4 13 28 154)(5 12 29 153)(6 11 30 152)(7 20 21 151)(8 19 22 160)(9 18 23 159)(10 17 24 158)(31 138 44 141)(32 137 45 150)(33 136 46 149)(34 135 47 148)(35 134 48 147)(36 133 49 146)(37 132 50 145)(38 131 41 144)(39 140 42 143)(40 139 43 142)(51 121 64 118)(52 130 65 117)(53 129 66 116)(54 128 67 115)(55 127 68 114)(56 126 69 113)(57 125 70 112)(58 124 61 111)(59 123 62 120)(60 122 63 119)(71 98 84 101)(72 97 85 110)(73 96 86 109)(74 95 87 108)(75 94 88 107)(76 93 89 106)(77 92 90 105)(78 91 81 104)(79 100 82 103)(80 99 83 102)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 31)(11 150)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(61 86)(62 87)(63 88)(64 89)(65 90)(66 81)(67 82)(68 83)(69 84)(70 85)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)
(1 130 30 112)(2 121 21 113)(3 122 22 114)(4 123 23 115)(5 124 24 116)(6 125 25 117)(7 126 26 118)(8 127 27 119)(9 128 28 120)(10 129 29 111)(11 70 157 52)(12 61 158 53)(13 62 159 54)(14 63 160 55)(15 64 151 56)(16 65 152 57)(17 66 153 58)(18 67 154 59)(19 68 155 60)(20 69 156 51)(31 109 49 91)(32 110 50 92)(33 101 41 93)(34 102 42 94)(35 103 43 95)(36 104 44 96)(37 105 45 97)(38 106 46 98)(39 107 47 99)(40 108 48 100)(71 144 89 136)(72 145 90 137)(73 146 81 138)(74 147 82 139)(75 148 83 140)(76 149 84 131)(77 150 85 132)(78 141 86 133)(79 142 87 134)(80 143 88 135)
(1 145 6 150)(2 146 7 141)(3 147 8 142)(4 148 9 143)(5 149 10 144)(11 37 16 32)(12 38 17 33)(13 39 18 34)(14 40 19 35)(15 31 20 36)(21 138 26 133)(22 139 27 134)(23 140 28 135)(24 131 29 136)(25 132 30 137)(41 158 46 153)(42 159 47 154)(43 160 48 155)(44 151 49 156)(45 152 50 157)(51 109 56 104)(52 110 57 105)(53 101 58 106)(54 102 59 107)(55 103 60 108)(61 93 66 98)(62 94 67 99)(63 95 68 100)(64 96 69 91)(65 97 70 92)(71 129 76 124)(72 130 77 125)(73 121 78 126)(74 122 79 127)(75 123 80 128)(81 113 86 118)(82 114 87 119)(83 115 88 120)(84 116 89 111)(85 117 90 112)
G:=sub<Sym(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), (1,16,25,157)(2,15,26,156)(3,14,27,155)(4,13,28,154)(5,12,29,153)(6,11,30,152)(7,20,21,151)(8,19,22,160)(9,18,23,159)(10,17,24,158)(31,138,44,141)(32,137,45,150)(33,136,46,149)(34,135,47,148)(35,134,48,147)(36,133,49,146)(37,132,50,145)(38,131,41,144)(39,140,42,143)(40,139,43,142)(51,121,64,118)(52,130,65,117)(53,129,66,116)(54,128,67,115)(55,127,68,114)(56,126,69,113)(57,125,70,112)(58,124,61,111)(59,123,62,120)(60,122,63,119)(71,98,84,101)(72,97,85,110)(73,96,86,109)(74,95,87,108)(75,94,88,107)(76,93,89,106)(77,92,90,105)(78,91,81,104)(79,100,82,103)(80,99,83,102), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,31)(11,150)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,86)(62,87)(63,88)(64,89)(65,90)(66,81)(67,82)(68,83)(69,84)(70,85)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)(137,152)(138,153)(139,154)(140,155), (1,130,30,112)(2,121,21,113)(3,122,22,114)(4,123,23,115)(5,124,24,116)(6,125,25,117)(7,126,26,118)(8,127,27,119)(9,128,28,120)(10,129,29,111)(11,70,157,52)(12,61,158,53)(13,62,159,54)(14,63,160,55)(15,64,151,56)(16,65,152,57)(17,66,153,58)(18,67,154,59)(19,68,155,60)(20,69,156,51)(31,109,49,91)(32,110,50,92)(33,101,41,93)(34,102,42,94)(35,103,43,95)(36,104,44,96)(37,105,45,97)(38,106,46,98)(39,107,47,99)(40,108,48,100)(71,144,89,136)(72,145,90,137)(73,146,81,138)(74,147,82,139)(75,148,83,140)(76,149,84,131)(77,150,85,132)(78,141,86,133)(79,142,87,134)(80,143,88,135), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,37,16,32)(12,38,17,33)(13,39,18,34)(14,40,19,35)(15,31,20,36)(21,138,26,133)(22,139,27,134)(23,140,28,135)(24,131,29,136)(25,132,30,137)(41,158,46,153)(42,159,47,154)(43,160,48,155)(44,151,49,156)(45,152,50,157)(51,109,56,104)(52,110,57,105)(53,101,58,106)(54,102,59,107)(55,103,60,108)(61,93,66,98)(62,94,67,99)(63,95,68,100)(64,96,69,91)(65,97,70,92)(71,129,76,124)(72,130,77,125)(73,121,78,126)(74,122,79,127)(75,123,80,128)(81,113,86,118)(82,114,87,119)(83,115,88,120)(84,116,89,111)(85,117,90,112)>;
G:=Group( (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), (1,16,25,157)(2,15,26,156)(3,14,27,155)(4,13,28,154)(5,12,29,153)(6,11,30,152)(7,20,21,151)(8,19,22,160)(9,18,23,159)(10,17,24,158)(31,138,44,141)(32,137,45,150)(33,136,46,149)(34,135,47,148)(35,134,48,147)(36,133,49,146)(37,132,50,145)(38,131,41,144)(39,140,42,143)(40,139,43,142)(51,121,64,118)(52,130,65,117)(53,129,66,116)(54,128,67,115)(55,127,68,114)(56,126,69,113)(57,125,70,112)(58,124,61,111)(59,123,62,120)(60,122,63,119)(71,98,84,101)(72,97,85,110)(73,96,86,109)(74,95,87,108)(75,94,88,107)(76,93,89,106)(77,92,90,105)(78,91,81,104)(79,100,82,103)(80,99,83,102), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,31)(11,150)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,86)(62,87)(63,88)(64,89)(65,90)(66,81)(67,82)(68,83)(69,84)(70,85)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)(137,152)(138,153)(139,154)(140,155), (1,130,30,112)(2,121,21,113)(3,122,22,114)(4,123,23,115)(5,124,24,116)(6,125,25,117)(7,126,26,118)(8,127,27,119)(9,128,28,120)(10,129,29,111)(11,70,157,52)(12,61,158,53)(13,62,159,54)(14,63,160,55)(15,64,151,56)(16,65,152,57)(17,66,153,58)(18,67,154,59)(19,68,155,60)(20,69,156,51)(31,109,49,91)(32,110,50,92)(33,101,41,93)(34,102,42,94)(35,103,43,95)(36,104,44,96)(37,105,45,97)(38,106,46,98)(39,107,47,99)(40,108,48,100)(71,144,89,136)(72,145,90,137)(73,146,81,138)(74,147,82,139)(75,148,83,140)(76,149,84,131)(77,150,85,132)(78,141,86,133)(79,142,87,134)(80,143,88,135), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,37,16,32)(12,38,17,33)(13,39,18,34)(14,40,19,35)(15,31,20,36)(21,138,26,133)(22,139,27,134)(23,140,28,135)(24,131,29,136)(25,132,30,137)(41,158,46,153)(42,159,47,154)(43,160,48,155)(44,151,49,156)(45,152,50,157)(51,109,56,104)(52,110,57,105)(53,101,58,106)(54,102,59,107)(55,103,60,108)(61,93,66,98)(62,94,67,99)(63,95,68,100)(64,96,69,91)(65,97,70,92)(71,129,76,124)(72,130,77,125)(73,121,78,126)(74,122,79,127)(75,123,80,128)(81,113,86,118)(82,114,87,119)(83,115,88,120)(84,116,89,111)(85,117,90,112) );
G=PermutationGroup([(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)], [(1,16,25,157),(2,15,26,156),(3,14,27,155),(4,13,28,154),(5,12,29,153),(6,11,30,152),(7,20,21,151),(8,19,22,160),(9,18,23,159),(10,17,24,158),(31,138,44,141),(32,137,45,150),(33,136,46,149),(34,135,47,148),(35,134,48,147),(36,133,49,146),(37,132,50,145),(38,131,41,144),(39,140,42,143),(40,139,43,142),(51,121,64,118),(52,130,65,117),(53,129,66,116),(54,128,67,115),(55,127,68,114),(56,126,69,113),(57,125,70,112),(58,124,61,111),(59,123,62,120),(60,122,63,119),(71,98,84,101),(72,97,85,110),(73,96,86,109),(74,95,87,108),(75,94,88,107),(76,93,89,106),(77,92,90,105),(78,91,81,104),(79,100,82,103),(80,99,83,102)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,31),(11,150),(12,141),(13,142),(14,143),(15,144),(16,145),(17,146),(18,147),(19,148),(20,149),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75),(61,86),(62,87),(63,88),(64,89),(65,90),(66,81),(67,82),(68,83),(69,84),(70,85),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(131,156),(132,157),(133,158),(134,159),(135,160),(136,151),(137,152),(138,153),(139,154),(140,155)], [(1,130,30,112),(2,121,21,113),(3,122,22,114),(4,123,23,115),(5,124,24,116),(6,125,25,117),(7,126,26,118),(8,127,27,119),(9,128,28,120),(10,129,29,111),(11,70,157,52),(12,61,158,53),(13,62,159,54),(14,63,160,55),(15,64,151,56),(16,65,152,57),(17,66,153,58),(18,67,154,59),(19,68,155,60),(20,69,156,51),(31,109,49,91),(32,110,50,92),(33,101,41,93),(34,102,42,94),(35,103,43,95),(36,104,44,96),(37,105,45,97),(38,106,46,98),(39,107,47,99),(40,108,48,100),(71,144,89,136),(72,145,90,137),(73,146,81,138),(74,147,82,139),(75,148,83,140),(76,149,84,131),(77,150,85,132),(78,141,86,133),(79,142,87,134),(80,143,88,135)], [(1,145,6,150),(2,146,7,141),(3,147,8,142),(4,148,9,143),(5,149,10,144),(11,37,16,32),(12,38,17,33),(13,39,18,34),(14,40,19,35),(15,31,20,36),(21,138,26,133),(22,139,27,134),(23,140,28,135),(24,131,29,136),(25,132,30,137),(41,158,46,153),(42,159,47,154),(43,160,48,155),(44,151,49,156),(45,152,50,157),(51,109,56,104),(52,110,57,105),(53,101,58,106),(54,102,59,107),(55,103,60,108),(61,93,66,98),(62,94,67,99),(63,95,68,100),(64,96,69,91),(65,97,70,92),(71,129,76,124),(72,130,77,125),(73,121,78,126),(74,122,79,127),(75,123,80,128),(81,113,86,118),(82,114,87,119),(83,115,88,120),(84,116,89,111),(85,117,90,112)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,0,0,0,0,40,40,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,39,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | C10.432+ (1+4) | C23.D10 | Dic5⋊4D4 | C4.Dic10 | D5×C4⋊C4 | C23.21D10 | C4×C5⋊D4 | D4×Dic5 | C23.18D10 | C20⋊2D4 | C5×C4⋊D4 | C4⋊D4 | C20 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{43}2_+^{(1+4)} % in TeX
G:=Group("C10.43ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1286);
// by ID
G=gap.SmallGroup(320,1286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,185,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=a^5*b^2,e^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^5*b^-1,b*d=d*b,e*b*e^-1=a^5*b,c*d=d*c,e*c*e^-1=a^5*c,e*d*e^-1=b^2*d>;
// generators/relations
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