C2xD10:3Q8 - GroupNames

G = C₂xD₁₀:₃Q₈ order 320 = 2⁶·5

Direct product of C₂ and D₁₀:₃Q₈

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂xD₁₀:₃Q₈, D₁₀:₄(C₂xQ₈), (C₂xQ₈):₂₈D₁₀, (C₂²xD₅):₇Q₈, (C₂²xQ₈):₅D₅, (C₂xC₂₀).₂₁₄D₄, C₂₀.₂₅₈(C₂xD₄), C₁₀:₅(C₂²:Q₈), C₂².₃₇(Q₈xD₅), C₄:Dic₅:₇₉C₂², (Q₈xC₁₀):₃₅C₂², C₁₀.₅₃(C₂²xQ₈), (C₂xC₁₀).₃₀₅C₂⁴, (C₂xC₂₀).₅₅₂C₂³, (C₂²xC₄).₃₈₄D₁₀, C₁₀.₁₅₃(C₂²xD₄), C₁₀.D₄:₇₅C₂², C₂³.₃₄₁(C₂²xD₅), C₂².₃₁₆(C₂³xD₅), (C₂²xC₂₀).₂₈₅C₂², (C₂²xC₁₀).₄₂₃C₂³, C₂².₃₉(Q₈:₂D₅), (C₂xDic₅).₁₅₇C₂³, (C₂³xD₅).₁₂₆C₂², (C₂²xD₅).₂₅₁C₂³, D₁₀:C₄.₁₅₇C₂², (C₂²xDic₅).₁₆₄C₂², (Q₈xC₂xC₁₀):₄C₂, C₅:₆(C₂xC₂²:Q₈), C₂.₃₅(C₂xQ₈xD₅), C₄.₉₈(C₂xC₅:D₄), (D₅xC₂²xC₄).₉C₂, (C₂xC₄:Dic₅):₄₆C₂, (C₂xC₁₀).₉₈(C₂xQ₈), C₁₀.₁₂₇(C₂xC₄oD₄), (C₂xC₁₀).₅₈₈(C₂xD₄), C₂.₃₄(C₂xQ₈:₂D₅), (C₂xC₄xD₅).₃₃₂C₂², C₂.₂₆(C₂²xC₅:D₄), (C₂xC₁₀.D₄):₄₉C₂, (C₂xC₄).₂₀₂(C₅:D₄), (C₂xC₄).₂₄₂(C₂²xD₅), C₂².₁₁₆(C₂xC₅:D₄), (C₂xD₁₀:C₄).₂₉C₂, (C₂xC₁₀).₂₀₀(C₄oD₄), SmallGroup(320,1485)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₂xD₁₀:₃Q₈

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂²xD₅ — C₂³xD₅ — D₅xC₂²xC₄ — C₂xD₁₀:₃Q₈

Lower central

C₅ — C₂xC₁₀ — C₂xD₁₀:₃Q₈

Upper central

C₁ — C₂³ — C₂²xQ₈

Generators and relations for C₂xD₁₀:₃Q₈
G = < a,b,c,d,e | a²=b¹⁰=c²=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=b⁵c, ce=ec, ede^-1=d^-1 >

Subgroups: 1038 in 322 conjugacy classes, 135 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₂²:C₄, C₂xC₄:C₄, C₂²:Q₈, C₂³xC₄, C₂²xQ₈, C₄xD₅, C₂xDic₅, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₅xQ₈, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂xC₂²:Q₈, C₁₀.D₄, C₄:Dic₅, D₁₀:C₄, C₂xC₄xD₅, C₂xC₄xD₅, C₂²xDic₅, C₂²xDic₅, C₂²xC₂₀, C₂²xC₂₀, Q₈xC₁₀, Q₈xC₁₀, C₂³xD₅, C₂xC₁₀.D₄, C₂xC₄:Dic₅, C₂xD₁₀:C₄, D₁₀:₃Q₈, D₅xC₂²xC₄, Q₈xC₂xC₁₀, C₂xD₁₀:₃Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₅, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, D₁₀, C₂²:Q₈, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₅:D₄, C₂²xD₅, C₂xC₂²:Q₈, Q₈xD₅, Q₈:₂D₅, C₂xC₅:D₄, C₂³xD₅, D₁₀:₃Q₈, C₂xQ₈xD₅, C₂xQ₈:₂D₅, C₂²xC₅:D₄, C₂xD₁₀:₃Q₈

Smallest permutation representation of C₂xD₁₀:₃Q₈
►On 160 points

Generators in S₁₆₀

(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 71)(19 72)(20 73)(21 100)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 122)(32 123)(33 124)(34 125)(35 126)(36 127)(37 128)(38 129)(39 130)(40 121)(41 120)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 141)(61 140)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(81 160)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)

(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)

(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 80)(20 79)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 100)(29 99)(30 98)(31 122)(32 121)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 112)(42 111)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 147)(52 146)(53 145)(54 144)(55 143)(56 142)(57 141)(58 150)(59 149)(60 148)(61 137)(62 136)(63 135)(64 134)(65 133)(66 132)(67 131)(68 140)(69 139)(70 138)(81 152)(82 151)(83 160)(84 159)(85 158)(86 157)(87 156)(88 155)(89 154)(90 153)

(1 45 25 34)(2 46 26 35)(3 47 27 36)(4 48 28 37)(5 49 29 38)(6 50 30 39)(7 41 21 40)(8 42 22 31)(9 43 23 32)(10 44 24 33)(11 134 154 145)(12 135 155 146)(13 136 156 147)(14 137 157 148)(15 138 158 149)(16 139 159 150)(17 140 160 141)(18 131 151 142)(19 132 152 143)(20 133 153 144)(51 71 62 82)(52 72 63 83)(53 73 64 84)(54 74 65 85)(55 75 66 86)(56 76 67 87)(57 77 68 88)(58 78 69 89)(59 79 70 90)(60 80 61 81)(91 122 102 111)(92 123 103 112)(93 124 104 113)(94 125 105 114)(95 126 106 115)(96 127 107 116)(97 128 108 117)(98 129 109 118)(99 130 110 119)(100 121 101 120)

(1 65 25 54)(2 66 26 55)(3 67 27 56)(4 68 28 57)(5 69 29 58)(6 70 30 59)(7 61 21 60)(8 62 22 51)(9 63 23 52)(10 64 24 53)(11 125 154 114)(12 126 155 115)(13 127 156 116)(14 128 157 117)(15 129 158 118)(16 130 159 119)(17 121 160 120)(18 122 151 111)(19 123 152 112)(20 124 153 113)(31 82 42 71)(32 83 43 72)(33 84 44 73)(34 85 45 74)(35 86 46 75)(36 87 47 76)(37 88 48 77)(38 89 49 78)(39 90 50 79)(40 81 41 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)

G:=sub<Sym(160)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,71)(19,72)(20,73)(21,100)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,121)(41,120)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,141)(61,140)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(81,160)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159), (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), (1,104)(2,103)(3,102)(4,101)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,80)(20,79)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,100)(29,99)(30,98)(31,122)(32,121)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,112)(42,111)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,147)(52,146)(53,145)(54,144)(55,143)(56,142)(57,141)(58,150)(59,149)(60,148)(61,137)(62,136)(63,135)(64,134)(65,133)(66,132)(67,131)(68,140)(69,139)(70,138)(81,152)(82,151)(83,160)(84,159)(85,158)(86,157)(87,156)(88,155)(89,154)(90,153), (1,45,25,34)(2,46,26,35)(3,47,27,36)(4,48,28,37)(5,49,29,38)(6,50,30,39)(7,41,21,40)(8,42,22,31)(9,43,23,32)(10,44,24,33)(11,134,154,145)(12,135,155,146)(13,136,156,147)(14,137,157,148)(15,138,158,149)(16,139,159,150)(17,140,160,141)(18,131,151,142)(19,132,152,143)(20,133,153,144)(51,71,62,82)(52,72,63,83)(53,73,64,84)(54,74,65,85)(55,75,66,86)(56,76,67,87)(57,77,68,88)(58,78,69,89)(59,79,70,90)(60,80,61,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,65,25,54)(2,66,26,55)(3,67,27,56)(4,68,28,57)(5,69,29,58)(6,70,30,59)(7,61,21,60)(8,62,22,51)(9,63,23,52)(10,64,24,53)(11,125,154,114)(12,126,155,115)(13,127,156,116)(14,128,157,117)(15,129,158,118)(16,130,159,119)(17,121,160,120)(18,122,151,111)(19,123,152,112)(20,124,153,113)(31,82,42,71)(32,83,43,72)(33,84,44,73)(34,85,45,74)(35,86,46,75)(36,87,47,76)(37,88,48,77)(38,89,49,78)(39,90,50,79)(40,81,41,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140)>;

G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,71)(19,72)(20,73)(21,100)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,121)(41,120)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,141)(61,140)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(81,160)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159), (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), (1,104)(2,103)(3,102)(4,101)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,80)(20,79)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,100)(29,99)(30,98)(31,122)(32,121)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,112)(42,111)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,147)(52,146)(53,145)(54,144)(55,143)(56,142)(57,141)(58,150)(59,149)(60,148)(61,137)(62,136)(63,135)(64,134)(65,133)(66,132)(67,131)(68,140)(69,139)(70,138)(81,152)(82,151)(83,160)(84,159)(85,158)(86,157)(87,156)(88,155)(89,154)(90,153), (1,45,25,34)(2,46,26,35)(3,47,27,36)(4,48,28,37)(5,49,29,38)(6,50,30,39)(7,41,21,40)(8,42,22,31)(9,43,23,32)(10,44,24,33)(11,134,154,145)(12,135,155,146)(13,136,156,147)(14,137,157,148)(15,138,158,149)(16,139,159,150)(17,140,160,141)(18,131,151,142)(19,132,152,143)(20,133,153,144)(51,71,62,82)(52,72,63,83)(53,73,64,84)(54,74,65,85)(55,75,66,86)(56,76,67,87)(57,77,68,88)(58,78,69,89)(59,79,70,90)(60,80,61,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,65,25,54)(2,66,26,55)(3,67,27,56)(4,68,28,57)(5,69,29,58)(6,70,30,59)(7,61,21,60)(8,62,22,51)(9,63,23,52)(10,64,24,53)(11,125,154,114)(12,126,155,115)(13,127,156,116)(14,128,157,117)(15,129,158,118)(16,130,159,119)(17,121,160,120)(18,122,151,111)(19,123,152,112)(20,124,153,113)(31,82,42,71)(32,83,43,72)(33,84,44,73)(34,85,45,74)(35,86,46,75)(36,87,47,76)(37,88,48,77)(38,89,49,78)(39,90,50,79)(40,81,41,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140) );

G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,101),(8,102),(9,103),(10,104),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,71),(19,72),(20,73),(21,100),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,122),(32,123),(33,124),(34,125),(35,126),(36,127),(37,128),(38,129),(39,130),(40,121),(41,120),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,148),(58,149),(59,150),(60,141),(61,140),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(81,160),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159)], [(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)], [(1,104),(2,103),(3,102),(4,101),(5,110),(6,109),(7,108),(8,107),(9,106),(10,105),(11,78),(12,77),(13,76),(14,75),(15,74),(16,73),(17,72),(18,71),(19,80),(20,79),(21,97),(22,96),(23,95),(24,94),(25,93),(26,92),(27,91),(28,100),(29,99),(30,98),(31,122),(32,121),(33,130),(34,129),(35,128),(36,127),(37,126),(38,125),(39,124),(40,123),(41,112),(42,111),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,147),(52,146),(53,145),(54,144),(55,143),(56,142),(57,141),(58,150),(59,149),(60,148),(61,137),(62,136),(63,135),(64,134),(65,133),(66,132),(67,131),(68,140),(69,139),(70,138),(81,152),(82,151),(83,160),(84,159),(85,158),(86,157),(87,156),(88,155),(89,154),(90,153)], [(1,45,25,34),(2,46,26,35),(3,47,27,36),(4,48,28,37),(5,49,29,38),(6,50,30,39),(7,41,21,40),(8,42,22,31),(9,43,23,32),(10,44,24,33),(11,134,154,145),(12,135,155,146),(13,136,156,147),(14,137,157,148),(15,138,158,149),(16,139,159,150),(17,140,160,141),(18,131,151,142),(19,132,152,143),(20,133,153,144),(51,71,62,82),(52,72,63,83),(53,73,64,84),(54,74,65,85),(55,75,66,86),(56,76,67,87),(57,77,68,88),(58,78,69,89),(59,79,70,90),(60,80,61,81),(91,122,102,111),(92,123,103,112),(93,124,104,113),(94,125,105,114),(95,126,106,115),(96,127,107,116),(97,128,108,117),(98,129,109,118),(99,130,110,119),(100,121,101,120)], [(1,65,25,54),(2,66,26,55),(3,67,27,56),(4,68,28,57),(5,69,29,58),(6,70,30,59),(7,61,21,60),(8,62,22,51),(9,63,23,52),(10,64,24,53),(11,125,154,114),(12,126,155,115),(13,127,156,116),(14,128,157,117),(15,129,158,118),(16,130,159,119),(17,121,160,120),(18,122,151,111),(19,123,152,112),(20,124,153,113),(31,82,42,71),(32,83,43,72),(33,84,44,73),(34,85,45,74),(35,86,46,75),(36,87,47,76),(37,88,48,77),(38,89,49,78),(39,90,50,79),(40,81,41,80),(91,142,102,131),(92,143,103,132),(93,144,104,133),(94,145,105,134),(95,146,106,135),(96,147,107,136),(97,148,108,137),(98,149,109,138),(99,150,110,139),(100,141,101,140)]])

68 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	5A	5B	10A	···	10N	20A	···	20X
order	1	2	···	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	5	5	10	···	10	20	···	20
size	1	1	···	1	10	10	10	10	2	2	2	2	4	4	4	4	10	10	10	10	20	20	20	20	2	2	2	···	2	4	···	4

68 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-	+		+	+		-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	D₅	C₄oD₄	D₁₀	D₁₀	C₅:D₄	Q₈xD₅	Q₈:₂D₅
kernel	C₂xD₁₀:₃Q₈	C₂xC₁₀.D₄	C₂xC₄:Dic₅	C₂xD₁₀:C₄	D₁₀:₃Q₈	D₅xC₂²xC₄	Q₈xC₂xC₁₀	C₂xC₂₀	C₂²xD₅	C₂²xQ₈	C₂xC₁₀	C₂²xC₄	C₂xQ₈	C₂xC₄	C₂²	C₂²
# reps	1	2	1	2	8	1	1	4	4	2	4	6	8	16	4	4

Matrix representation of C₂xD₁₀:₃Q₈ ►in GL₆(F₄₁)

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	7	0	0	0	0
34	34	0	0	0	0
0	0	0	7	0	0
0	0	35	35	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	34	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	36	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

24	1	0	0	0	0
40	17	0	0	0	0
0	0	23	40	0	0
0	0	36	18	0	0
0	0	0	0	0	1
0	0	0	0	40	0

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	0	32
0	0	0	0	32	0

G:=sub<GL(6,GF(41))| [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],[1,34,0,0,0,0,7,34,0,0,0,0,0,0,0,35,0,0,0,0,7,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,34,1,0,0,0,0,0,0,40,36,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,36,0,0,0,0,40,18,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0] >;

C₂xD₁₀:₃Q₈ in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes_3Q_8

% in TeX

G:=Group("C2xD10:3Q8");

// GroupNames label

G:=SmallGroup(320,1485);

// by ID

G=gap.SmallGroup(320,1485);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2xD10:3Q8 order 320 = 26·5

Direct product of C2 and D10:3Q8

G = C₂xD₁₀:₃Q₈ order 320 = 2⁶·5

Direct product of C₂ and D₁₀:₃Q₈