C2^2xQ8:D5 - GroupNames

G = C₂²×Q₈⋊D₅ order 320 = 2⁶·5

Direct product of C₂² and Q₈⋊D₅

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

Aliases: C₂²×Q₈⋊D₅, C₂₀.₃₀C₂⁴, D₂₀.₂₇C₂³, (C₂×Q₈)⋊₂₆D₁₀, C₁₀⋊₄(C₂×SD₁₆), Q₈⋊₄(C₂²×D₅), (C₅×Q₈)⋊₄C₂³, (C₂²×Q₈)⋊₃D₅, C₅⋊₄(C₂²×SD₁₆), C₅⋊₂C₈⋊₁₀C₂³, (C₂×C₁₀)⋊₁₅SD₁₆, C₂₀.₂₅₄(C₂×D₄), (C₂×C₂₀).₂₁₀D₄, C₄.₃₀(C₂³×D₅), (Q₈×C₁₀)⋊₃₃C₂², (C₂×C₂₀).₅₄₇C₂³, (C₂²×D₂₀).₁₉C₂, (C₂²×C₁₀).₂₀₉D₄, (C₂²×C₄).₃₈₁D₁₀, C₁₀.₁₄₉(C₂²×D₄), (C₂×D₂₀).₂₈₄C₂², C₂³.₁₀₆(C₅⋊D₄), (C₂²×C₂₀).₂₇₉C₂², (Q₈×C₂×C₁₀)⋊₂C₂, C₄.₂₄(C₂×C₅⋊D₄), (C₂×C₅⋊₂C₈)⋊₄₀C₂², (C₂²×C₅⋊₂C₈)⋊₁₃C₂, (C₂×C₁₀).₅₈₄(C₂×D₄), C₂.₂₂(C₂²×C₅⋊D₄), (C₂×C₄).₁₅₄(C₅⋊D₄), (C₂×C₄).₆₂₉(C₂²×D₅), C₂².₁₁₂(C₂×C₅⋊D₄), SmallGroup(320,1479)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₂²×Q₈⋊D₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — D₂₀ — C₂×D₂₀ — C₂²×D₂₀ — C₂²×Q₈⋊D₅

Lower central

C₅ — C₁₀ — C₂₀ — C₂²×Q₈⋊D₅

Upper central

C₁ — C₂³ — C₂²×C₄ — C₂²×Q₈

Generators and relations for C₂²×Q₈⋊D₅
G = < a,b,c,d,e,f | a²=b²=c⁴=e⁵=f²=1, d²=c², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd^-1=fcf=c^-1, ce=ec, de=ed, fdf=c^-1d, fef=e^-1 >

Subgroups: 1182 in 298 conjugacy classes, 127 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂₀, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂²×C₈, C₂×SD₁₆, C₂²×D₄, C₂²×Q₈, C₅⋊₂C₈, D₂₀, D₂₀, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, C₂²×SD₁₆, C₂×C₅⋊₂C₈, Q₈⋊D₅, C₂×D₂₀, C₂×D₂₀, C₂²×C₂₀, C₂²×C₂₀, Q₈×C₁₀, Q₈×C₁₀, C₂³×D₅, C₂²×C₅⋊₂C₈, C₂×Q₈⋊D₅, C₂²×D₂₀, Q₈×C₂×C₁₀, C₂²×Q₈⋊D₅
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, SD₁₆, C₂×D₄, C₂⁴, D₁₀, C₂×SD₁₆, C₂²×D₄, C₅⋊D₄, C₂²×D₅, C₂²×SD₁₆, Q₈⋊D₅, C₂×C₅⋊D₄, C₂³×D₅, C₂×Q₈⋊D₅, C₂²×C₅⋊D₄, C₂²×Q₈⋊D₅

Smallest permutation representation of C₂²×Q₈⋊D₅
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	8A	···	8H	10A	···	10N	20A	···	20X
order	1	2	···	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	8	···	8	10	···	10	20	···	20
size	1	1	···	1	20	20	20	20	2	2	2	2	4	4	4	4	2	2	10	···	10	2	···	2	4	···	4

68 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

SD₁₆

D₁₀

C₅⋊D₄

Q₈⋊D₅

kernel

C₂²×Q₈⋊D₅

C₂²×C₅⋊₂C₈

C₂×Q₈⋊D₅

C₂²×D₂₀

Q₈×C₂×C₁₀

C₂×C₂₀

C₂²×C₁₀

C₂²×Q₈

C₂×C₁₀

C₂²×C₄

C₂×Q₈

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂²×Q₈⋊D₅ ►in GL₆(𝔽₄₁)

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	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	1	0
0	0	0	0	0	1

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	39
0	0	0	0	1	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	0	11
0	0	0	0	26	0

34	40	0	0	0	0
1	0	0	0	0	0
0	0	35	40	0	0
0	0	36	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

34	40	0	0	0	0
7	7	0	0	0	0
0	0	0	34	0	0
0	0	35	0	0	0
0	0	0	0	1	0
0	0	0	0	40	40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,35,36,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,0,35,0,0,0,0,34,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;

C₂²×Q₈⋊D₅ in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes D_5

% in TeX

G:=Group("C2^2xQ8:D5");

// GroupNames label

G:=SmallGroup(320,1479);

// by ID

G=gap.SmallGroup(320,1479);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,136,1684,235,102,12550]);

// Polycyclic

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

// generators/relations

G = C22×Q8⋊D5 order 320 = 26·5

Direct product of C22 and Q8⋊D5

G = C₂²×Q₈⋊D₅ order 320 = 2⁶·5

Direct product of C₂² and Q₈⋊D₅