C2^3xC5:D4 - GroupNames

G = C₂³×C₅⋊D₄ order 320 = 2⁶·5

Direct product of C₂³ and C₅⋊D₄

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

Aliases: C₂³×C₅⋊D₄, C₂⁵⋊₃D₅, D₁₀⋊₃C₂⁴, C₂⁴⋊₁₃D₁₀, C₁₀.₂₀C₂⁵, Dic₅⋊₂C₂⁴, C₅⋊₃(D₄×C₂³), (C₂×C₁₀)⋊₃C₂⁴, (C₂⁴×C₁₀)⋊₄C₂, (D₅×C₂⁴)⋊₆C₂, C₁₀⋊₃(C₂²×D₄), (C₂²×C₁₀)⋊₁₉D₄, C₂.₂₀(D₅×C₂⁴), C₂³⋊₆(C₂²×D₅), C₂²⋊₂(C₂³×D₅), (C₂²×C₁₀)⋊₉C₂³, (C₂²×D₅)⋊₉C₂³, (C₂³×C₁₀)⋊₁₉C₂², (C₂³×Dic₅)⋊₁₂C₂, (C₂×Dic₅)⋊₁₃C₂³, (C₂³×D₅)⋊₂₄C₂², (C₂²×Dic₅)⋊₅₄C₂², (C₂×C₁₀)⋊₁₇(C₂×D₄), SmallGroup(320,1627)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂³×C₅⋊D₄

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂³×D₅ — D₅×C₂⁴ — C₂³×C₅⋊D₄

Lower central

C₅ — C₁₀ — C₂³×C₅⋊D₄

Upper central

C₁ — C₂⁴ — C₂⁵

Generators and relations for C₂³×C₅⋊D₄
G = < a,b,c,d,e,f | a²=b²=c²=d⁵=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 4126 in 1362 conjugacy classes, 543 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, Dic₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³×C₄, C₂²×D₄, C₂⁵, C₂⁵, C₂×Dic₅, C₅⋊D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, D₄×C₂³, C₂²×Dic₅, C₂×C₅⋊D₄, C₂³×D₅, C₂³×D₅, C₂³×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₂³×Dic₅, C₂²×C₅⋊D₄, D₅×C₂⁴, C₂⁴×C₁₀, C₂³×C₅⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, C₂⁵, C₅⋊D₄, C₂²×D₅, D₄×C₂³, C₂×C₅⋊D₄, C₂³×D₅, C₂²×C₅⋊D₄, D₅×C₂⁴, C₂³×C₅⋊D₄

Smallest permutation representation of C₂³×C₅⋊D₄
►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 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)

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

(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)

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

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

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

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

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

104 conjugacy classes

class	1	2A	···	2O	2P	···	2W	2X	···	2AE	4A	···	4H	5A	5B	10A	···	10BJ
order	1	2	···	2	2	···	2	2	···	2	4	···	4	5	5	10	···	10
size	1	1	···	1	2	···	2	10	···	10	10	···	10	2	2	2	···	2

104 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₅	D₁₀	C₅⋊D₄
kernel	C₂³×C₅⋊D₄	C₂³×Dic₅	C₂²×C₅⋊D₄	D₅×C₂⁴	C₂⁴×C₁₀	C₂²×C₁₀	C₂⁵	C₂⁴	C₂³
# reps	1	1	28	1	1	8	2	30	32

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

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

40	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	40	0
0	0	0	0	0	40

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

40	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	34	40	0	0
0	0	0	0	24	1
0	0	0	0	38	17

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

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,24,38,0,0,0,0,1,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

C₂³×C₅⋊D₄ in GAP, Magma, Sage, TeX

C_2^3\times C_5\rtimes D_4

% in TeX

G:=Group("C2^3xC5:D4");

// GroupNames label

G:=SmallGroup(320,1627);

// by ID

G=gap.SmallGroup(320,1627);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23×C5⋊D4 order 320 = 26·5

Direct product of C23 and C5⋊D4

G = C₂³×C₅⋊D₄ order 320 = 2⁶·5

Direct product of C₂³ and C₅⋊D₄