C2xC4^2:2D5 - GroupNames

G = C₂×C₄²⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂ and C₄²⋊₂D₅

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

Aliases: C₂×C₄²⋊₂D₅, C₄²⋊₃₆D₁₀, (C₂×C₄²)⋊₄D₅, (C₄×C₂₀)⋊₄₇C₂², (C₂×C₁₀).₂₂C₂⁴, C₁₀⋊₁(C₄²⋊₂C₂), (C₂×C₂₀).₆₉₅C₂³, (C₂²×C₄).₄₀₉D₁₀, (C₂×Dic₅).₆C₂³, (C₂²×D₅).₄C₂³, C₂².₆₅(C₂³×D₅), C₂².₇₀(C₄○D₂₀), C₁₀.D₄⋊₄₁C₂², (C₂³×D₅).₂₉C₂², C₂³.₃₁₇(C₂²×D₅), D₁₀⋊C₄.₈₁C₂², (C₂²×C₁₀).₃₈₄C₂³, (C₂²×C₂₀).₅₀₅C₂², (C₂²×Dic₅).₇₅C₂², (C₂×C₄×C₂₀)⋊₄C₂, C₅⋊₁(C₂×C₄²⋊₂C₂), C₁₀.₉(C₂×C₄○D₄), C₂.₁₁(C₂×C₄○D₂₀), (C₂×C₁₀).₉₈(C₄○D₄), (C₂×C₁₀.D₄)⋊₁₆C₂, (C₂×C₄).₆₅₀(C₂²×D₅), (C₂×D₁₀⋊C₄).₁₈C₂, SmallGroup(320,1150)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₄²⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂×D₁₀⋊C₄ — C₂×C₄²⋊₂D₅

Lower central

C₅ — C₂×C₁₀ — C₂×C₄²⋊₂D₅

Upper central

C₁ — C₂³ — C₂×C₄²

Generators and relations for C₂×C₄²⋊₂D₅
G = < a,b,c,d,e | a²=b⁴=c⁴=d⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc², cd=dc, ece=b²c^-1, ede=d^-1 >

Subgroups: 846 in 246 conjugacy classes, 111 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂×C₄²⋊₂C₂, C₁₀.D₄, D₁₀⋊C₄, C₄×C₂₀, C₂²×Dic₅, C₂²×C₂₀, C₂³×D₅, C₄²⋊₂D₅, C₂×C₁₀.D₄, C₂×D₁₀⋊C₄, C₂×C₄×C₂₀, C₂×C₄²⋊₂D₅
Quotients: C₁, C₂, C₂², C₂³, D₅, C₄○D₄, C₂⁴, D₁₀, C₄²⋊₂C₂, C₂×C₄○D₄, C₂²×D₅, C₂×C₄²⋊₂C₂, C₄○D₂₀, C₂³×D₅, C₄²⋊₂D₅, C₂×C₄○D₂₀, C₂×C₄²⋊₂D₅

Smallest permutation representation of C₂×C₄²⋊₂D₅
►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)

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

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

(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)

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

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

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

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

92 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4L	4M	···	4R	5A	5B	10A	···	10N	20A	···	20AV
order	1	2	···	2	2	2	4	···	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	20	20	2	···	2	20	···	20	2	2	2	···	2	2	···	2

92 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2
type	+	+	+	+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₂	D₅	C₄○D₄	D₁₀	D₁₀	C₄○D₂₀
kernel	C₂×C₄²⋊₂D₅	C₄²⋊₂D₅	C₂×C₁₀.D₄	C₂×D₁₀⋊C₄	C₂×C₄×C₂₀	C₂×C₄²	C₂×C₁₀	C₄²	C₂²×C₄	C₂²
# reps	1	8	3	3	1	2	12	8	6	48

Matrix representation of C₂×C₄²⋊₂D₅ ►in GL₆(𝔽₄₁)

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	0
0	0	0	0	0	40

9	0	0	0	0	0
0	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	11	9
0	0	0	0	32	30

17	1	0	0	0	0
40	24	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	32	0
0	0	0	0	0	32

0	40	0	0	0	0
1	34	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

7	40	0	0	0	0
7	34	0	0	0	0
0	0	34	40	0	0
0	0	7	7	0	0
0	0	0	0	34	40
0	0	0	0	7	7

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,32,0,0,0,0,9,30],[17,40,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,40,34,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[7,7,0,0,0,0,40,34,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,40,7] >;

C₂×C₄²⋊₂D₅ in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_2D_5

% in TeX

G:=Group("C2xC4^2:2D5");

// GroupNames label

G:=SmallGroup(320,1150);

// by ID

G=gap.SmallGroup(320,1150);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,1571,136,12550]);

// Polycyclic

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

// generators/relations

G = C2×C42⋊2D5 order 320 = 26·5

Direct product of C2 and C42⋊2D5

G = C₂×C₄²⋊₂D₅ order 320 = 2⁶·5

Direct product of C₂ and C₄²⋊₂D₅