(C5xD4).D4 - GroupNames

G = (C₅×D₄).D₄ order 320 = 2⁶·5

8^th non-split extension by C₅×D₄ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₅×D₄).₈D₄, (C₂×SD₁₆)⋊₈D₅, (D₄×Dic₅)⋊₇C₂, C₂₀.₁₇₂(C₂×D₄), (C₂×C₈).₁₄₅D₁₀, D₄.₃(C₅⋊D₄), C₅⋊₇(D₄.₂D₄), (C₂×Q₈).₅₁D₁₀, D₂₀⋊₅C₄⋊₃₄C₂, (C₁₀×SD₁₆)⋊₁₉C₂, (C₂×D₄).₁₄₅D₁₀, C₂₀.₉₉(C₄○D₄), C₁₀.₆₀(C₄○D₈), Q₈⋊Dic₅⋊₂₇C₂, C₂₀.₂₃D₄⋊₃C₂, C₂₀.₈Q₈⋊₃₄C₂, (C₂×Dic₅).₇₆D₄, C₂².₂₆₂(D₄×D₅), C₄.₁₁(D₄⋊₂D₅), C₂.₂₇(D₄₀⋊C₂), C₁₀.₇₇(C₈⋊C₂²), (C₂×C₄₀).₂₉₂C₂², (C₂×C₂₀).₄₄₂C₂³, (D₄×C₁₀).₉₁C₂², (Q₈×C₁₀).₇₂C₂², C₁₀.₁₁₄(C₄⋊D₄), (C₂×D₂₀).₁₂₂C₂², C₄⋊Dic₅.₁₇₂C₂², (C₄×Dic₅).₅₅C₂², C₂.₂₆(Dic₅⋊D₄), C₂.₂₆(SD₁₆⋊₃D₅), (C₂×D₄⋊D₅).₈C₂, C₄.₄₀(C₂×C₅⋊D₄), (C₂×C₁₀).₃₅₄(C₂×D₄), (C₂×C₄).₅₃₁(C₂²×D₅), (C₂×C₅⋊₂C₈).₁₅₄C₂², SmallGroup(320,792)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₅×D₄).D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄×Dic₅ — D₄×Dic₅ — (C₅×D₄).D₄

Lower central

C₅ — C₁₀ — C₂×C₂₀ — (C₅×D₄).D₄

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for (C₅×D₄).D₄
G = < a,b,c,d,e | a⁵=b⁴=c²=d⁴=e²=1, ab=ba, ac=ca, dad^-1=eae=a^-1, cbc=ebe=b^-1, bd=db, cd=dc, ece=b^-1c, ede=b²d^-1 >

Subgroups: 534 in 124 conjugacy classes, 41 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄×D₄, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₅⋊₂C₈, C₄₀, D₂₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, D₄.₂D₄, C₂×C₅⋊₂C₈, C₄×Dic₅, C₄⋊Dic₅, D₁₀⋊C₄, D₄⋊D₅, C₂³.D₅, C₂×C₄₀, C₅×SD₁₆, C₂×D₂₀, C₂²×Dic₅, D₄×C₁₀, Q₈×C₁₀, C₂₀.₈Q₈, D₂₀⋊₅C₄, Q₈⋊Dic₅, C₂×D₄⋊D₅, D₄×Dic₅, C₂₀.₂₃D₄, C₁₀×SD₁₆, (C₅×D₄).D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₄○D₄, D₁₀, C₄⋊D₄, C₄○D₈, C₈⋊C₂², C₅⋊D₄, C₂²×D₅, D₄.₂D₄, D₄×D₅, D₄⋊₂D₅, C₂×C₅⋊D₄, D₄₀⋊C₂, SD₁₆⋊₃D₅, Dic₅⋊D₄, (C₅×D₄).D₄

Smallest permutation representation of (C₅×D₄).D₄
►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 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(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)

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

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

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

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

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

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

47 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	8A	8B	8C	8D	10A	···	10F	10G	10H	10I	10J	20A	20B	20C	20D	20E	20F	20G	20H	40A	···	40H
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	8	8	8	8	10	···	10	10	10	10	10	20	20	20	20	20	20	20	20	40	···	40
size	1	1	1	1	4	4	40	2	2	8	10	10	20	20	20	2	2	4	4	20	20	2	···	2	8	8	8	8	4	4	4	4	8	8	8	8	4	···	4

47 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+	+			+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₅	C₄○D₄	D₁₀	D₁₀	D₁₀	C₄○D₈	C₅⋊D₄	C₈⋊C₂²	D₄⋊₂D₅	D₄×D₅	D₄₀⋊C₂	SD₁₆⋊₃D₅
kernel	(C₅×D₄).D₄	C₂₀.₈Q₈	D₂₀⋊₅C₄	Q₈⋊Dic₅	C₂×D₄⋊D₅	D₄×Dic₅	C₂₀.₂₃D₄	C₁₀×SD₁₆	C₂×Dic₅	C₅×D₄	C₂×SD₁₆	C₂₀	C₂×C₈	C₂×D₄	C₂×Q₈	C₁₀	D₄	C₁₀	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	8	1	2	2	4	4

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

0	40	0	0	0	0
1	6	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	1	5
0	0	0	0	16	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	24	19
0	0	0	0	28	17

6	35	0	0	0	0
40	35	0	0	0	0
0	0	0	40	0	0
0	0	1	0	0	0
0	0	0	0	9	0
0	0	0	0	0	9

6	35	0	0	0	0
40	35	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	16	40

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,5,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,28,0,0,0,0,19,17],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;

(C₅×D₄).D₄ in GAP, Magma, Sage, TeX

(C_5\times D_4).D_4

% in TeX

G:=Group("(C5xD4).D4");

// GroupNames label

G:=SmallGroup(320,792);

// by ID

G=gap.SmallGroup(320,792);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1094,135,184,570,297,136,12550]);

// Polycyclic

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

// generators/relations

G = (C5×D4).D4 order 320 = 26·5

8th non-split extension by C5×D4 of D4 acting via D4/C2=C22

G = (C₅×D₄).D₄ order 320 = 2⁶·5

8^th non-split extension by C₅×D₄ of D₄ acting via D₄/C₂=C₂²