C2^2xQ8:2D5 - GroupNames

G = C₂²×Q₈⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂² and Q₈⋊₂D₅

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

Aliases: C₂²×Q₈⋊₂D₅, C₁₀.₉C₂⁵, D₂₀⋊₁₀C₂³, C₂₀.₄₄C₂⁴, D₁₀.₄C₂⁴, Dic₅.₂₆C₂⁴, (C₂×Q₈)⋊₃₇D₁₀, (C₄×D₅)⋊₆C₂³, Q₈⋊₆(C₂²×D₅), (C₅×Q₈)⋊₇C₂³, (C₂²×Q₈)⋊₁₁D₅, C₂.₁₀(D₅×C₂⁴), C₄.₄₄(C₂³×D₅), (C₂²×D₂₀)⋊₂₃C₂, (C₂×D₂₀)⋊₆₂C₂², (Q₈×C₁₀)⋊₄₃C₂², (C₂×C₂₀).₅₆₅C₂³, (C₂×C₁₀).₃₂₉C₂⁴, (C₂²×C₄).₃₉₁D₁₀, C₂².₅₅(C₂³×D₅), C₂³.₃₅₀(C₂²×D₅), (C₂²×C₂₀).₃₀₁C₂², (C₂²×C₁₀).₄₃₆C₂³, (C₂×Dic₅).₃₈₂C₂³, (C₂²×D₅).₂₅₇C₂³, (C₂³×D₅).₁₃₀C₂², (C₂²×Dic₅).₂₉₁C₂², C₁₀⋊₃(C₂×C₄○D₄), (Q₈×C₂×C₁₀)⋊₁₀C₂, C₅⋊₃(C₂²×C₄○D₄), (D₅×C₂²×C₄)⋊₁₀C₂, (C₂×C₄×D₅)⋊₆₀C₂², (C₂×C₁₀)⋊₁₉(C₄○D₄), (C₂×C₄).₆₄₅(C₂²×D₅), SmallGroup(320,1616)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂²×Q₈⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂³×D₅ — D₅×C₂²×C₄ — C₂²×Q₈⋊₂D₅

Lower central

C₅ — C₁₀ — C₂²×Q₈⋊₂D₅

Upper central

C₁ — C₂³ — C₂²×Q₈

Generators and relations for C₂²×Q₈⋊₂D₅
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, df=fd, fef=e^-1 >

Subgroups: 2718 in 890 conjugacy classes, 463 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₄×D₅, D₂₀, C₂×Dic₅, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₄○D₄, C₂×C₄×D₅, C₂×D₂₀, Q₈⋊₂D₅, C₂²×Dic₅, C₂²×C₂₀, Q₈×C₁₀, C₂³×D₅, D₅×C₂²×C₄, C₂²×D₂₀, C₂×Q₈⋊₂D₅, Q₈×C₂×C₁₀, C₂²×Q₈⋊₂D₅
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄○D₄, C₂⁴, D₁₀, C₂×C₄○D₄, C₂⁵, C₂²×D₅, C₂²×C₄○D₄, Q₈⋊₂D₅, C₂³×D₅, C₂×Q₈⋊₂D₅, D₅×C₂⁴, C₂²×Q₈⋊₂D₅

Smallest permutation representation of C₂²×Q₈⋊₂D₅
►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 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(81 126)(82 127)(83 128)(84 129)(85 130)(86 121)(87 122)(88 123)(89 124)(90 125)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)

(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)

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

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

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

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

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

80 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4L	4M	···	4T	5A	5B	10A	···	10N	20A	···	20X
order	1	2	···	2	2	···	2	4	···	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	10	···	10	2	···	2	5	···	5	2	2	2	···	2	4	···	4

80 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₅	C₄○D₄	D₁₀	D₁₀	Q₈⋊₂D₅
kernel	C₂²×Q₈⋊₂D₅	D₅×C₂²×C₄	C₂²×D₂₀	C₂×Q₈⋊₂D₅	Q₈×C₂×C₁₀	C₂²×Q₈	C₂×C₁₀	C₂²×C₄	C₂×Q₈	C₂²
# reps	1	3	3	24	1	2	8	6	24	8

Matrix representation of C₂²×Q₈⋊₂D₅ ►in GL₇(𝔽₄₁)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	40	0	0	0
0	0	0	0	40	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

40	0	0	0	0	0	0
0	0	1	0	0	0	0
0	40	0	0	0	0	0
0	0	0	0	1	0	0
0	0	0	40	0	0	0
0	0	0	0	0	40	0
0	0	0	0	0	0	40

1	0	0	0	0	0	0
0	9	0	0	0	0	0
0	0	32	0	0	0	0
0	0	0	32	0	0	0
0	0	0	0	9	0	0
0	0	0	0	0	40	0
0	0	0	0	0	0	40

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

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

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,35,36,0,0,0,0,0,40,40],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,0,0,0,0,0,34,0] >;

C₂²×Q₈⋊₂D₅ in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes_2D_5

% in TeX

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

// GroupNames label

G:=SmallGroup(320,1616);

// by ID

G=gap.SmallGroup(320,1616);

# by ID

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

// 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,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C22×Q8⋊2D5 order 320 = 26·5

Direct product of C22 and Q8⋊2D5

G = C₂²×Q₈⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂² and Q₈⋊₂D₅