C5xC2^3:2D4 - GroupNames

G = C₅×C₂³⋊₂D₄ order 320 = 2⁶·5

Direct product of C₅ and C₂³⋊₂D₄

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C₅×C₂³⋊₂D₄, (C₂×C₂₀)⋊₂₄D₄, C₂³⋊₂(C₅×D₄), (C₂²×C₁₀)⋊₅D₄, (C₂²×D₄)⋊₁C₁₀, C₂⁴.₄(C₂×C₁₀), C₁₀.₉₀C₂²≀C₂, C₂².₆₈(D₄×C₁₀), C₁₀.₄₀(C₄⋊₁D₄), C₂.C₄²⋊₉C₁₀, (C₂³×C₁₀).₄C₂², C₁₀.₁₃₇(C₄⋊D₄), C₂³.₇₅(C₂²×C₁₀), (C₂²×C₁₀).₄₅₆C₂³, (C₂²×C₂₀).₄₀₀C₂², (C₂×C₄)⋊₂(C₅×D₄), (D₄×C₂×C₁₀)⋊₁₃C₂, C₂.₆(C₅×C₄⋊D₄), C₂.₃(C₅×C₄⋊₁D₄), (C₂×C₂²⋊C₄)⋊₆C₁₀, C₂.₄(C₅×C₂²≀C₂), (C₁₀×C₂²⋊C₄)⋊₂₇C₂, (C₂×C₁₀).₆₀₈(C₂×D₄), (C₂²×C₄).₄(C₂×C₁₀), C₂².₃₅(C₅×C₄○D₄), (C₂×C₁₀).₂₁₆(C₄○D₄), (C₅×C₂.C₄²)⋊₂₅C₂, SmallGroup(320,893)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₅×C₂³⋊₂D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₁₀ — C₂³×C₁₀ — D₄×C₂×C₁₀ — C₅×C₂³⋊₂D₄

Lower central

C₁ — C₂³ — C₅×C₂³⋊₂D₄

Upper central

C₁ — C₂²×C₁₀ — C₅×C₂³⋊₂D₄

Generators and relations for C₅×C₂³⋊₂D₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe^-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 634 in 322 conjugacy classes, 86 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂²×D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊₂D₄, C₅×C₂²⋊C₄, C₂²×C₂₀, C₂²×C₂₀, D₄×C₁₀, C₂³×C₁₀, C₅×C₂.C₄², C₁₀×C₂²⋊C₄, D₄×C₂×C₁₀, C₅×C₂³⋊₂D₄
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₄○D₄, C₂×C₁₀, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₅×D₄, C₂²×C₁₀, C₂³⋊₂D₄, D₄×C₁₀, C₅×C₄○D₄, C₅×C₂²≀C₂, C₅×C₄⋊D₄, C₅×C₄⋊₁D₄, C₅×C₂³⋊₂D₄

Smallest permutation representation of C₅×C₂³⋊₂D₄
►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)

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

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

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

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

(6 152)(7 153)(8 154)(9 155)(10 151)(16 144)(17 145)(18 141)(19 142)(20 143)(21 160)(22 156)(23 157)(24 158)(25 159)(31 136)(32 137)(33 138)(34 139)(35 140)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 133)(97 134)(98 135)(99 131)(100 132)(101 123)(102 124)(103 125)(104 121)(105 122)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)

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

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

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

110 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4H	5A	5B	5C	5D	10A	···	10AB	10AC	···	10AZ	20A	···	20AF
order	1	2	···	2	2	···	2	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	4	···	4	4	···	4	1	1	1	1	1	···	1	4	···	4	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+					+	+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	D₄	C₄○D₄	C₅×D₄	C₅×D₄	C₅×C₄○D₄
kernel	C₅×C₂³⋊₂D₄	C₅×C₂.C₄²	C₁₀×C₂²⋊C₄	D₄×C₂×C₁₀	C₂³⋊₂D₄	C₂.C₄²	C₂×C₂²⋊C₄	C₂²×D₄	C₂×C₂₀	C₂²×C₁₀	C₂×C₁₀	C₂×C₄	C₂³	C₂²
# reps	1	1	3	3	4	4	12	12	6	6	2	24	24	8

Matrix representation of C₅×C₂³⋊₂D₄ ►in GL₆(𝔽₄₁)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	37	0	0	0
0	0	0	37	0	0
0	0	0	0	18	0
0	0	0	0	0	18

1	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	34	39
0	0	0	0	24	7

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	0
0	0	0	0	0	40

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

0	1	0	0	0	0
40	0	0	0	0	0
0	0	37	1	0	0
0	0	24	4	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	8	40	0	0
0	0	0	0	1	0
0	0	0	0	34	40

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,24,0,0,0,0,39,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,24,0,0,0,0,1,4,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

C₅×C₂³⋊₂D₄ in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes_2D_4

% in TeX

G:=Group("C5xC2^3:2D4");

// GroupNames label

G:=SmallGroup(320,893);

// by ID

G=gap.SmallGroup(320,893);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1731]);

// Polycyclic

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

// generators/relations

G = C5×C23⋊2D4 order 320 = 26·5

Direct product of C5 and C23⋊2D4

G = C₅×C₂³⋊₂D₄ order 320 = 2⁶·5

Direct product of C₅ and C₂³⋊₂D₄