C2^2xQ8xD5 - GroupNames

G = C₂²×Q₈×D₅ order 320 = 2⁶·5

Direct product of C₂², Q₈ and D₅

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

Aliases: C₂²×Q₈×D₅, C₁₀.₈C₂⁵, C₂₀.₄₃C₂⁴, D₁₀.₂₃C₂⁴, Dic₅.₄C₂⁴, Dic₁₀⋊₁₀C₂³, C₅⋊₂(Q₈×C₂³), (C₅×Q₈)⋊₆C₂³, C₁₀⋊₂(C₂²×Q₈), C₂.₉(D₅×C₂⁴), C₄.₄₃(C₂³×D₅), (Q₈×C₁₀)⋊₄₂C₂², (C₄×D₅).₆₈C₂³, (C₂×C₂₀).₅₆₄C₂³, (C₂×C₁₀).₃₂₈C₂⁴, (C₂²×C₄).₃₉₀D₁₀, C₂².₅₄(C₂³×D₅), (C₂×Dic₁₀)⋊₇₃C₂², (C₂²×Dic₁₀)⋊₂₄C₂, C₂³.₃₄₉(C₂²×D₅), (C₂²×C₂₀).₃₀₀C₂², (C₂²×C₁₀).₄₃₅C₂³, (C₂×Dic₅).₃₀₇C₂³, (C₂²×D₅).₂₉₉C₂³, (C₂³×D₅).₁₄₆C₂², (C₂²×Dic₅).₂₆₃C₂², (Q₈×C₂×C₁₀)⋊₉C₂, (C₂×C₁₀)⋊₁₀(C₂×Q₈), (D₅×C₂²×C₄).₁₀C₂, (C₂×C₄×D₅).₃₃₄C₂², (C₂×C₄).₆₄₄(C₂²×D₅), SmallGroup(320,1615)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂²×Q₈×D₅

Chief series

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

Lower central

C₅ — C₁₀ — C₂²×Q₈×D₅

Upper central

C₁ — C₂³ — 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=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2238 in 850 conjugacy classes, 503 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, Dic₁₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, Q₈×C₂³, C₂×Dic₁₀, C₂×C₄×D₅, Q₈×D₅, C₂²×Dic₅, C₂²×C₂₀, Q₈×C₁₀, C₂³×D₅, C₂²×Dic₁₀, D₅×C₂²×C₄, C₂×Q₈×D₅, Q₈×C₂×C₁₀, C₂²×Q₈×D₅
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, C₂⁴, D₁₀, C₂²×Q₈, C₂⁵, C₂²×D₅, Q₈×C₂³, Q₈×D₅, C₂³×D₅, C₂×Q₈×D₅, D₅×C₂⁴, C₂²×Q₈×D₅

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

Generators in S₁₆₀

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4L	4M	···	4X	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	5	···	5	2	···	2	10	···	10	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	Q₈	D₅	D₁₀	D₁₀	Q₈×D₅
kernel	C₂²×Q₈×D₅	C₂²×Dic₁₀	D₅×C₂²×C₄	C₂×Q₈×D₅	Q₈×C₂×C₁₀	C₂²×D₅	C₂²×Q₈	C₂²×C₄	C₂×Q₈	C₂²
# reps	1	3	3	24	1	8	2	6	24	8

Matrix representation of C₂²×Q₈×D₅ ►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	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	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	0	1
0	0	0	0	40	0

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	9	0
0	0	0	0	0	32

40	1	0	0	0	0
5	35	0	0	0	0
0	0	34	1	0	0
0	0	40	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	0	0	0	0	0
5	1	0	0	0	0
0	0	40	7	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂²×Q₈×D₅ in GAP, Magma, Sage, TeX

C_2^2\times Q_8\times D_5

% in TeX

G:=Group("C2^2xQ8xD5");

// GroupNames label

G:=SmallGroup(320,1615);

// by ID

G=gap.SmallGroup(320,1615);

# by ID

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

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

// generators/relations

G = C22×Q8×D5 order 320 = 26·5

Direct product of C22, Q8 and D5

G = C₂²×Q₈×D₅ order 320 = 2⁶·5

Direct product of C₂², Q₈ and D₅