D4xC2^2xC10 - GroupNames

G = D₄×C₂²×C₁₀ order 320 = 2⁶·5

Direct product of C₂²×C₁₀ and D₄

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

Aliases: D₄×C₂²×C₁₀, C₂₀⋊₄C₂⁴, C₂⁵⋊₃C₁₀, C₁₀.₂₁C₂⁵, C₄⋊(C₂³×C₁₀), (C₂³×C₄)⋊₇C₁₀, (C₂⁴×C₁₀)⋊₂C₂, (C₂×C₁₀)⋊₂C₂⁴, C₂⁴⋊₈(C₂×C₁₀), C₂²⋊(C₂³×C₁₀), (C₂³×C₂₀)⋊₁₆C₂, (C₂×C₂₀)⋊₁₇C₂³, C₂.₁(C₂⁴×C₁₀), C₂³⋊₃(C₂²×C₁₀), (C₂²×C₁₀)⋊₈C₂³, (C₂²×C₂₀)⋊₆₆C₂², (C₂³×C₁₀)⋊₁₇C₂², (C₂×C₄)⋊₄(C₂²×C₁₀), (C₂²×C₄)⋊₁₉(C₂×C₁₀), SmallGroup(320,1629)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — D₄×C₂²×C₁₀

Chief series

C₁ — C₂ — C₁₀ — C₂×C₁₀ — C₅×D₄ — D₄×C₁₀ — D₄×C₂×C₁₀ — D₄×C₂²×C₁₀

Lower central

C₁ — C₂ — D₄×C₂²×C₁₀

Upper central

C₁ — C₂³×C₁₀ — D₄×C₂²×C₁₀

Generators and relations for D₄×C₂²×C₁₀
G = < a,b,c,d,e | a²=b²=c¹⁰=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 1874 in 1362 conjugacy classes, 850 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂³×C₄, C₂²×D₄, C₂⁵, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, D₄×C₂³, C₂²×C₂₀, D₄×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₂³×C₁₀, C₂³×C₂₀, D₄×C₂×C₁₀, C₂⁴×C₁₀, D₄×C₂²×C₁₀
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₂²×D₄, C₂⁵, C₅×D₄, C₂²×C₁₀, D₄×C₂³, D₄×C₁₀, C₂³×C₁₀, D₄×C₂×C₁₀, C₂⁴×C₁₀, D₄×C₂²×C₁₀

Smallest permutation representation of D₄×C₂²×C₁₀
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

200 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	4A	···	4H	5A	5B	5C	5D	10A	···	10BH	10BI	···	10DT	20A	···	20AF
order	1	2	···	2	2	···	2	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	2	···	2	2	···	2	1	1	1	1	1	···	1	2	···	2	2	···	2

200 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	C₅×D₄
kernel	D₄×C₂²×C₁₀	C₂³×C₂₀	D₄×C₂×C₁₀	C₂⁴×C₁₀	D₄×C₂³	C₂³×C₄	C₂²×D₄	C₂⁵	C₂²×C₁₀	C₂³
# reps	1	1	28	2	4	4	112	8	8	32

Matrix representation of D₄×C₂²×C₁₀ ►in GL₅(𝔽₄₁)

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

40	0	0	0	0
0	40	0	0	0
0	0	40	0	0
0	0	0	40	0
0	0	0	0	40

1	0	0	0	0
0	1	0	0	0
0	0	31	0	0
0	0	0	40	0
0	0	0	0	40

40	0	0	0	0
0	40	0	0	0
0	0	40	0	0
0	0	0	40	2
0	0	0	40	1

40	0	0	0	0
0	40	0	0	0
0	0	40	0	0
0	0	0	1	39
0	0	0	0	40

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,40,0,0,0,2,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,39,40] >;

D₄×C₂²×C₁₀ in GAP, Magma, Sage, TeX

D_4\times C_2^2\times C_{10}

% in TeX

G:=Group("D4xC2^2xC10");

// GroupNames label

G:=SmallGroup(320,1629);

// by ID

G=gap.SmallGroup(320,1629);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = D4×C22×C10 order 320 = 26·5

Direct product of C22×C10 and D4

G = D₄×C₂²×C₁₀ order 320 = 2⁶·5

Direct product of C₂²×C₁₀ and D₄