C2^2xD5:C8 - GroupNames

G = C₂²×D₅⋊C₈ order 320 = 2⁶·5

Direct product of C₂² and D₅⋊C₈

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

Aliases: C₂²×D₅⋊C₈, Dic₅.₁₄C₂⁴, C₅⋊C₈⋊₄C₂³, C₅⋊₁(C₂³×C₈), D₅⋊(C₂²×C₈), D₁₀⋊₁₁(C₂×C₈), C₁₀⋊₁(C₂²×C₈), (C₂²×D₅)⋊₇C₈, C₂.₁(C₂³×F₅), C₁₀.₁(C₂³×C₄), C₄.₅₇(C₂²×F₅), C₂³.₆₄(C₂×F₅), (C₂²×C₄).₂₈F₅, (C₂²×C₂₀).₃₆C₄, C₂₀.₉₇(C₂²×C₄), (C₂³×D₅).₁₇C₄, (C₄×D₅).₉₀C₂³, D₁₀.₄₃(C₂²×C₄), C₂².₅₄(C₂²×F₅), Dic₅.₄₃(C₂²×C₄), (C₂×Dic₅).₃₆₁C₂³, (C₂²×Dic₅).₂₈₁C₂², (C₂×C₁₀)⋊₄(C₂×C₈), (C₂×C₄×D₅).₄₇C₄, (C₂²×C₅⋊C₈)⋊₁₁C₂, (C₂×C₅⋊C₈)⋊₁₄C₂², (C₄×D₅).₉₆(C₂×C₄), (C₂×C₄).₁₇₂(C₂×F₅), (D₅×C₂²×C₄).₃₆C₂, (C₂×C₂₀).₁₈₀(C₂×C₄), (C₂×C₄×D₅).₄₁₅C₂², (C₂²×C₁₀).₇₇(C₂×C₄), (C₂×C₁₀).₉₅(C₂²×C₄), (C₂×Dic₅).₁₉₇(C₂×C₄), (C₂²×D₅).₁₃₁(C₂×C₄), SmallGroup(320,1587)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂²×D₅⋊C₈

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂²×C₅⋊C₈ — C₂²×D₅⋊C₈

Lower central

C₅ — C₂²×D₅⋊C₈

Upper central

C₁ — C₂²×C₄

Generators and relations for C₂²×D₅⋊C₈
G = < a,b,c,d,e | a²=b²=c⁵=d²=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c^-1, ece^-1=c³, ede^-1=c²d >

Subgroups: 906 in 338 conjugacy classes, 196 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂²×C₈, C₂³×C₄, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, C₂³×C₈, D₅⋊C₈, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂²×Dic₅, C₂²×C₂₀, C₂³×D₅, C₂×D₅⋊C₈, C₂²×C₅⋊C₈, D₅×C₂²×C₄, C₂²×D₅⋊C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, C₂⁴, F₅, C₂²×C₈, C₂³×C₄, C₂×F₅, C₂³×C₈, D₅⋊C₈, C₂²×F₅, C₂×D₅⋊C₈, C₂³×F₅, C₂²×D₅⋊C₈

Smallest permutation representation of C₂²×D₅⋊C₈
►On 160 points

Generators in S₁₆₀

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

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

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

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

(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)

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

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

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

80 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4H	4I	···	4P	5	8A	···	8AF	10A	···	10G	20A	···	20H
order	1	2	···	2	2	···	2	4	···	4	4	···	4	5	8	···	8	10	···	10	20	···	20
size	1	1	···	1	5	···	5	1	···	1	5	···	5	4	5	···	5	4	···	4	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4
type	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₈	F₅	C₂×F₅	C₂×F₅	D₅⋊C₈
kernel	C₂²×D₅⋊C₈	C₂×D₅⋊C₈	C₂²×C₅⋊C₈	D₅×C₂²×C₄	C₂×C₄×D₅	C₂²×C₂₀	C₂³×D₅	C₂²×D₅	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	12	2	1	12	2	2	32	1	6	1	8

Matrix representation of C₂²×D₅⋊C₈ ►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

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

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

1	0	0	0	0	0
0	38	0	0	0	0
0	0	2	39	38	0
0	0	40	39	0	2
0	0	2	0	39	40
0	0	0	38	39	2

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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40],[1,0,0,0,0,0,0,38,0,0,0,0,0,0,2,40,2,0,0,0,39,39,0,38,0,0,38,0,39,39,0,0,0,2,40,2] >;

C₂²×D₅⋊C₈ in GAP, Magma, Sage, TeX

C_2^2\times D_5\rtimes C_8

% in TeX

G:=Group("C2^2xD5:C8");

// GroupNames label

G:=SmallGroup(320,1587);

// by ID

G=gap.SmallGroup(320,1587);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,136,102,6278,818]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^2=e^8=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,d*c*d=c^-1,e*c*e^-1=c^3,e*d*e^-1=c^2*d>;

// generators/relations

G = C22×D5⋊C8 order 320 = 26·5

Direct product of C22 and D5⋊C8

G = C₂²×D₅⋊C₈ order 320 = 2⁶·5

Direct product of C₂² and D₅⋊C₈