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G = C5×D4.2D4order 320 = 26·5

Direct product of C5 and D4.2D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×D4.2D4, C4⋊C85C10, (C4×D4)⋊4C10, D4.2(C5×D4), (D4×C20)⋊33C2, (C10×D8).9C2, (C5×D4).27D4, (C2×D8).2C10, C4.35(D4×C10), Q8⋊C46C10, C20.396(C2×D4), (C2×C20).325D4, C4.4D43C10, D4⋊C411C10, (C10×SD16)⋊29C2, (C2×SD16)⋊12C10, C42.18(C2×C10), C22.87(D4×C10), C20.345(C4○D4), C10.121(C4○D8), (C2×C40).301C22, (C4×C20).260C22, (C2×C20).922C23, C10.146(C4⋊D4), C10.136(C8⋊C22), (D4×C10).187C22, (Q8×C10).161C22, (C5×C4⋊C8)⋊24C2, C2.8(C5×C4○D8), C4.44(C5×C4○D4), (C2×C4).30(C5×D4), C4⋊C4.55(C2×C10), (C2×C8).38(C2×C10), C2.15(C5×C4⋊D4), C2.11(C5×C8⋊C22), (C2×Q8).6(C2×C10), (C5×D4⋊C4)⋊35C2, (C5×Q8⋊C4)⋊29C2, (C2×D4).57(C2×C10), (C2×C10).643(C2×D4), (C5×C4.4D4)⋊23C2, (C5×C4⋊C4).376C22, (C2×C4).97(C22×C10), SmallGroup(320,964)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×D4.2D4
C1C2C4C2×C4C2×C20D4×C10C5×C4.4D4 — C5×D4.2D4
C1C2C2×C4 — C5×D4.2D4
C1C2×C10C4×C20 — C5×D4.2D4

Generators and relations for C5×D4.2D4
 G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 250 in 124 conjugacy classes, 54 normal (50 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×2], D4 [×3], Q8 [×2], C23 [×2], C10 [×3], C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×7], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40 [×2], C2×C20 [×3], C2×C20 [×4], C5×D4 [×2], C5×D4 [×3], C5×Q8 [×2], C22×C10 [×2], D4.2D4, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4, C2×C40 [×2], C5×D8 [×2], C5×SD16 [×2], C22×C20, D4×C10 [×2], Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊C8, D4×C20, C5×C4.4D4, C10×D8, C10×SD16, C5×D4.2D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C4○D8, C8⋊C22, C5×D4 [×4], C22×C10, D4.2D4, D4×C10 [×2], C5×C4○D4, C5×C4⋊D4, C5×C4○D8, C5×C8⋊C22, C5×D4.2D4

Smallest permutation representation of C5×D4.2D4
On 160 points
Generators in S160
(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 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 127 160 18)(7 128 156 19)(8 129 157 20)(9 130 158 16)(10 126 159 17)(21 35 142 151)(22 31 143 152)(23 32 144 153)(24 33 145 154)(25 34 141 155)(26 56 42 51)(27 57 43 52)(28 58 44 53)(29 59 45 54)(30 60 41 55)(46 79 91 108)(47 80 92 109)(48 76 93 110)(49 77 94 106)(50 78 95 107)(61 75 83 97)(62 71 84 98)(63 72 85 99)(64 73 81 100)(65 74 82 96)(86 148 131 119)(87 149 132 120)(88 150 133 116)(89 146 134 117)(90 147 135 118)(101 137 123 115)(102 138 124 111)(103 139 125 112)(104 140 121 113)(105 136 122 114)
(1 77)(2 78)(3 79)(4 80)(5 76)(6 87)(7 88)(8 89)(9 90)(10 86)(11 110)(12 106)(13 107)(14 108)(15 109)(16 147)(17 148)(18 149)(19 150)(20 146)(21 138)(22 139)(23 140)(24 136)(25 137)(26 96)(27 97)(28 98)(29 99)(30 100)(31 103)(32 104)(33 105)(34 101)(35 102)(36 48)(37 49)(38 50)(39 46)(40 47)(41 73)(42 74)(43 75)(44 71)(45 72)(51 65)(52 61)(53 62)(54 63)(55 64)(56 82)(57 83)(58 84)(59 85)(60 81)(66 94)(67 95)(68 91)(69 92)(70 93)(111 142)(112 143)(113 144)(114 145)(115 141)(116 128)(117 129)(118 130)(119 126)(120 127)(121 153)(122 154)(123 155)(124 151)(125 152)(131 159)(132 160)(133 156)(134 157)(135 158)
(1 121 41 134)(2 122 42 135)(3 123 43 131)(4 124 44 132)(5 125 45 133)(6 109 35 98)(7 110 31 99)(8 106 32 100)(9 107 33 96)(10 108 34 97)(11 103 29 88)(12 104 30 89)(13 105 26 90)(14 101 27 86)(15 102 28 87)(16 95 24 82)(17 91 25 83)(18 92 21 84)(19 93 22 85)(20 94 23 81)(36 112 54 116)(37 113 55 117)(38 114 51 118)(39 115 52 119)(40 111 53 120)(46 141 61 126)(47 142 62 127)(48 143 63 128)(49 144 64 129)(50 145 65 130)(56 147 67 136)(57 148 68 137)(58 149 69 138)(59 150 70 139)(60 146 66 140)(71 160 80 151)(72 156 76 152)(73 157 77 153)(74 158 78 154)(75 159 79 155)
(1 89 12 134)(2 90 13 135)(3 86 14 131)(4 87 15 132)(5 88 11 133)(6 92 160 47)(7 93 156 48)(8 94 157 49)(9 95 158 50)(10 91 159 46)(16 107 130 78)(17 108 126 79)(18 109 127 80)(19 110 128 76)(20 106 129 77)(21 98 142 71)(22 99 143 72)(23 100 144 73)(24 96 145 74)(25 97 141 75)(26 122 42 105)(27 123 43 101)(28 124 44 102)(29 125 45 103)(30 121 41 104)(31 85 152 63)(32 81 153 64)(33 82 154 65)(34 83 155 61)(35 84 151 62)(36 116 70 150)(37 117 66 146)(38 118 67 147)(39 119 68 148)(40 120 69 149)(51 114 56 136)(52 115 57 137)(53 111 58 138)(54 112 59 139)(55 113 60 140)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B5C5D8A8B8C8D10A···10L10M···10T10U10V10W10X20A···20P20Q···20AB20AC20AD20AE20AF40A···40P
order1222222444444445555888810···1010···101010101020···2020···202020202040···40
size111144822224448111144441···14···488882···24···488884···4

95 irreducible representations

dim11111111111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4D4C4○D4C4○D8C5×D4C5×D4C5×C4○D4C5×C4○D8C8⋊C22C5×C8⋊C22
kernelC5×D4.2D4C5×D4⋊C4C5×Q8⋊C4C5×C4⋊C8D4×C20C5×C4.4D4C10×D8C10×SD16D4.2D4D4⋊C4Q8⋊C4C4⋊C8C4×D4C4.4D4C2×D8C2×SD16C2×C20C5×D4C20C10C2×C4D4C4C2C10C2
# reps111111114444444422248881614

Matrix representation of C5×D4.2D4 in GL4(𝔽41) generated by

16000
01600
00160
00016
,
40000
04000
0001
00400
,
04000
40000
001229
002929
,
0900
9000
00320
00032
,
03200
9000
00320
0009
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29],[0,9,0,0,9,0,0,0,0,0,32,0,0,0,0,32],[0,9,0,0,32,0,0,0,0,0,32,0,0,0,0,9] >;

C5×D4.2D4 in GAP, Magma, Sage, TeX

C_5\times D_4._2D_4
% in TeX

G:=Group("C5xD4.2D4");
// GroupNames label

G:=SmallGroup(320,964);
// by ID

G=gap.SmallGroup(320,964);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,10085,2539,124]);
// Polycyclic

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

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