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G = C4×D4⋊D5order 320 = 26·5

Direct product of C4 and D4⋊D5

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

Aliases: C4×D4⋊D5, C208D8, C42.207D10, C55(C4×D8), D43(C4×D5), (C4×D4)⋊1D5, (D4×C20)⋊1C2, (C4×D20)⋊19C2, D2020(C2×C4), C10.99(C4×D4), C10.51(C2×D8), C4⋊C4.241D10, (C2×C20).253D4, D206C444C2, (C2×D4).188D10, C20.48(C4○D4), C10.88(C4○D8), C4.36(C4○D20), D4⋊Dic543C2, C10.D845C2, C20.56(C22×C4), (C4×C20).84C22, (C2×C20).335C23, C2.3(D4.8D10), (C2×D20).244C22, (D4×C10).230C22, C4⋊Dic5.326C22, C4.21(C2×C4×D5), (C4×C52C8)⋊8C2, C2.3(C2×D4⋊D5), C52C820(C2×C4), (C5×D4)⋊18(C2×C4), C2.15(C4×C5⋊D4), (C2×D4⋊D5).10C2, (C2×C10).466(C2×D4), C22.75(C2×C5⋊D4), (C2×C4).101(C5⋊D4), (C5×C4⋊C4).272C22, (C2×C4).435(C22×D5), (C2×C52C8).252C22, SmallGroup(320,640)

Series: Derived Chief Lower central Upper central

C1C20 — C4×D4⋊D5
C1C5C10C2×C10C2×C20C2×D20C2×D4⋊D5 — C4×D4⋊D5
C5C10C20 — C4×D4⋊D5
C1C2×C4C42C4×D4

Generators and relations for C4×D4⋊D5
 G = < a,b,c,d,e | a4=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 502 in 134 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4, C2×D4, Dic5, C20 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4, C4×D4, C2×D8, C52C8 [×2], C52C8, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5, C22×C10, C4×D8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×C20, D4×C10, C4×C52C8, C10.D8, D206C4, D4⋊Dic5, C4×D20, C2×D4⋊D5, D4×C20, C4×D4⋊D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4×D8, D4⋊D5 [×2], C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×D4⋊D5, D4.8D10, C4×D4⋊D5

Smallest permutation representation of C4×D4⋊D5
On 160 points
Generators in S160
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 141 101 121)(82 142 102 122)(83 143 103 123)(84 144 104 124)(85 145 105 125)(86 146 106 126)(87 147 107 127)(88 148 108 128)(89 149 109 129)(90 150 110 130)(91 151 111 131)(92 152 112 132)(93 153 113 133)(94 154 114 134)(95 155 115 135)(96 156 116 136)(97 157 117 137)(98 158 118 138)(99 159 119 139)(100 160 120 140)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 156)(42 157)(43 158)(44 159)(45 160)(46 151)(47 152)(48 153)(49 154)(50 155)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 65)(42 64)(43 63)(44 62)(45 61)(46 70)(47 69)(48 68)(49 67)(50 66)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(81 115)(82 114)(83 113)(84 112)(85 111)(86 120)(87 119)(88 118)(89 117)(90 116)(91 105)(92 104)(93 103)(94 102)(95 101)(96 110)(97 109)(98 108)(99 107)(100 106)(121 155)(122 154)(123 153)(124 152)(125 151)(126 160)(127 159)(128 158)(129 157)(130 156)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10F10G···10N20A···20H20I···20X
order12222222444444444444558···810···1010···1020···2020···20
size1111442020111122224420202210···102···24···42···24···4

68 irreducible representations

dim1111111112222222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5D8C4○D4D10D10D10C4○D8C5⋊D4C4×D5C4○D20D4⋊D5D4.8D10
kernelC4×D4⋊D5C4×C52C8C10.D8D206C4D4⋊Dic5C4×D20C2×D4⋊D5D4×C20D4⋊D5C2×C20C4×D4C20C20C42C4⋊C4C2×D4C10C2×C4D4C4C4C2
# reps1111111182242222488844

Matrix representation of C4×D4⋊D5 in GL4(𝔽41) generated by

9000
0900
00400
00040
,
40000
04000
00137
002140
,
17100
402400
00034
00350
,
04000
13400
0010
0001
,
34100
34700
0010
002140
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[17,40,0,0,1,24,0,0,0,0,0,35,0,0,34,0],[0,1,0,0,40,34,0,0,0,0,1,0,0,0,0,1],[34,34,0,0,1,7,0,0,0,0,1,21,0,0,0,40] >;

C4×D4⋊D5 in GAP, Magma, Sage, TeX

C_4\times D_4\rtimes D_5
% in TeX

G:=Group("C4xD4:D5");
// GroupNames label

G:=SmallGroup(320,640);
// by ID

G=gap.SmallGroup(320,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,58,1684,851,102,12550]);
// Polycyclic

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

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