C2^3xD20 - GroupNames

G = C₂³×D₂₀ order 320 = 2⁶·5

Direct product of C₂³ and D₂₀

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

Aliases: C₂³×D₂₀, C₂₀⋊₂C₂⁴, D₁₀⋊₁C₂⁴, C₁₀.₃C₂⁵, C₂⁴.₈₂D₁₀, C₅⋊₁(D₄×C₂³), (C₂³×C₄)⋊₇D₅, C₄⋊₂(C₂³×D₅), (C₂³×C₂₀)⋊₉C₂, (D₅×C₂⁴)⋊₄C₂, C₁₀⋊₁(C₂²×D₄), C₂.₄(D₅×C₂⁴), (C₂×C₂₀)⋊₁₄C₂³, (C₂²×C₁₀)⋊₁₆D₄, (C₂²×C₄)⋊₄₅D₁₀, (C₂²×D₅)⋊₇C₂³, (C₂×C₁₀).₃₂₅C₂⁴, (C₂²×C₂₀)⋊₆₁C₂², (C₂³×D₅)⋊₂₂C₂², C₂².₅₃(C₂³×D₅), C₂³.₃₄₆(C₂²×D₅), (C₂²×C₁₀).₄₃₂C₂³, (C₂³×C₁₀).₁₁₅C₂², (C₂×C₁₀)⋊₁₂(C₂×D₄), (C₂×C₄)⋊₁₁(C₂²×D₅), SmallGroup(320,1610)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂³×D₂₀

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂³×D₅ — D₅×C₂⁴ — C₂³×D₂₀

Lower central

C₅ — C₁₀ — C₂³×D₂₀

Upper central

C₁ — C₂⁴ — C₂³×C₄

Generators and relations for C₂³×D₂₀
G = < a,b,c,d,e | a²=b²=c²=d²⁰=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 5150 in 1362 conjugacy classes, 543 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂³×C₄, C₂²×D₄, C₂⁵, D₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, D₄×C₂³, C₂×D₂₀, C₂²×C₂₀, C₂³×D₅, C₂³×D₅, C₂³×C₁₀, C₂²×D₂₀, C₂³×C₂₀, D₅×C₂⁴, C₂³×D₂₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, C₂⁵, D₂₀, C₂²×D₅, D₄×C₂³, C₂×D₂₀, C₂³×D₅, C₂²×D₂₀, D₅×C₂⁴, C₂³×D₂₀

Smallest permutation representation of C₂³×D₂₀
►On 160 points

Generators in S₁₆₀

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

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

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

(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)

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

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

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

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

104 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	4A	···	4H	5A	5B	10A	···	10AD	20A	···	20AF
order	1	2	···	2	2	···	2	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	10	···	10	2	···	2	2	2	2	···	2	2	···	2

104 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	D₄	D₅	D₁₀	D₁₀	D₂₀
kernel	C₂³×D₂₀	C₂²×D₂₀	C₂³×C₂₀	D₅×C₂⁴	C₂²×C₁₀	C₂³×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	28	1	2	8	2	28	2	32

Matrix representation of C₂³×D₂₀ ►in GL₆(𝔽₄₁)

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

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	1	0	0	0
0	0	0	1	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	1	40	0	0
0	0	8	34	0	0
0	0	0	0	28	39
0	0	0	0	2	16

40	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	33	1	0	0
0	0	0	0	0	40
0	0	0	0	40	0

G:=sub<GL(6,GF(41))| [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],[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],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,28,2,0,0,0,0,39,16],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

C₂³×D₂₀ in GAP, Magma, Sage, TeX

C_2^3\times D_{20}

% in TeX

G:=Group("C2^3xD20");

// GroupNames label

G:=SmallGroup(320,1610);

// by ID

G=gap.SmallGroup(320,1610);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1684,102,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=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 = C23×D20 order 320 = 26·5

Direct product of C23 and D20

G = C₂³×D₂₀ order 320 = 2⁶·5

Direct product of C₂³ and D₂₀