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G = C2.D8⋊D5order 320 = 26·5

5th semidirect product of C2.D8 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.D85D5, C4⋊C4.50D10, (C2×C8).29D10, C4⋊D20.9C2, D205C427C2, D101C826C2, D206C422C2, C20.40(C4○D4), C4.82(C4○D20), C10.76(C4○D8), C20.Q821C2, (C22×D5).36D4, C22.231(D4×D5), C2.23(D8⋊D5), C10.42(C8⋊C22), (C2×C40).243C22, (C2×C20).301C23, C4.30(Q82D5), (C2×Dic5).224D4, (C2×D20).87C22, C55(C23.19D4), C2.14(Q8.D10), C4⋊Dic5.126C22, C2.17(D10.13D4), C10.47(C22.D4), C4⋊C47D57C2, (C5×C2.D8)⋊13C2, (C2×C4×D5).44C22, (C2×C10).306(C2×D4), (C5×C4⋊C4).94C22, (C2×C52C8).71C22, (C2×C4).404(C22×D5), SmallGroup(320,512)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C2.D8⋊D5
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C2.D8⋊D5
C5C10C2×C20 — C2.D8⋊D5
C1C22C2×C4C2.D8

Generators and relations for C2.D8⋊D5
 G = < a,b,c,d,e | a2=b8=d5=e2=1, c2=a, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab5, cd=dc, ede=d-1 >

Subgroups: 526 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×D20, C2×D20, C20.Q8, D206C4, D101C8, D205C4, C5×C2.D8, C4⋊C47D5, C4⋊D20, C2.D8⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, Q82D5, D10.13D4, D8⋊D5, Q8.D10, C2.D8⋊D5

Smallest permutation representation of C2.D8⋊D5
On 160 points
Generators in S160
(1 72)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 137)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 89)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 118)(42 119)(43 120)(44 113)(45 114)(46 115)(47 116)(48 117)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 97)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 121)(80 122)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(145 157)(146 158)(147 159)(148 160)(149 153)(150 154)(151 155)(152 156)
(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)
(1 152 72 156)(2 151 65 155)(3 150 66 154)(4 149 67 153)(5 148 68 160)(6 147 69 159)(7 146 70 158)(8 145 71 157)(9 127 138 77)(10 126 139 76)(11 125 140 75)(12 124 141 74)(13 123 142 73)(14 122 143 80)(15 121 144 79)(16 128 137 78)(17 46 90 115)(18 45 91 114)(19 44 92 113)(20 43 93 120)(21 42 94 119)(22 41 95 118)(23 48 96 117)(24 47 89 116)(25 105 36 135)(26 112 37 134)(27 111 38 133)(28 110 39 132)(29 109 40 131)(30 108 33 130)(31 107 34 129)(32 106 35 136)(49 87 98 59)(50 86 99 58)(51 85 100 57)(52 84 101 64)(53 83 102 63)(54 82 103 62)(55 81 104 61)(56 88 97 60)
(1 61 19 37 75)(2 62 20 38 76)(3 63 21 39 77)(4 64 22 40 78)(5 57 23 33 79)(6 58 24 34 80)(7 59 17 35 73)(8 60 18 36 74)(9 150 53 42 132)(10 151 54 43 133)(11 152 55 44 134)(12 145 56 45 135)(13 146 49 46 136)(14 147 50 47 129)(15 148 51 48 130)(16 149 52 41 131)(25 124 71 88 91)(26 125 72 81 92)(27 126 65 82 93)(28 127 66 83 94)(29 128 67 84 95)(30 121 68 85 96)(31 122 69 86 89)(32 123 70 87 90)(97 114 105 141 157)(98 115 106 142 158)(99 116 107 143 159)(100 117 108 144 160)(101 118 109 137 153)(102 119 110 138 154)(103 120 111 139 155)(104 113 112 140 156)
(1 142)(2 10)(3 144)(4 12)(5 138)(6 14)(7 140)(8 16)(9 68)(11 70)(13 72)(15 66)(17 113)(18 41)(19 115)(20 43)(21 117)(22 45)(23 119)(24 47)(25 101)(26 49)(27 103)(28 51)(29 97)(30 53)(31 99)(32 55)(33 102)(34 50)(35 104)(36 52)(37 98)(38 54)(39 100)(40 56)(42 96)(44 90)(46 92)(48 94)(57 110)(58 129)(59 112)(60 131)(61 106)(62 133)(63 108)(64 135)(65 139)(67 141)(69 143)(71 137)(73 156)(74 149)(75 158)(76 151)(77 160)(78 145)(79 154)(80 147)(81 136)(82 111)(83 130)(84 105)(85 132)(86 107)(87 134)(88 109)(89 116)(91 118)(93 120)(95 114)(121 150)(122 159)(123 152)(124 153)(125 146)(126 155)(127 148)(128 157)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120402244810102020224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8C4○D20C8⋊C22Q82D5D4×D5D8⋊D5Q8.D10
kernelC2.D8⋊D5C20.Q8D206C4D101C8D205C4C5×C2.D8C4⋊C47D5C4⋊D20C2×Dic5C22×D5C2.D8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111111124424812244

Matrix representation of C2.D8⋊D5 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
0053300
0033600
00001229
00001212
,
4000000
0400000
00373100
0014400
000090
0000032
,
010000
40340000
001000
000100
000010
000001
,
0400000
4000000
00373100
0022400
000009
0000320

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,5,3,0,0,0,0,33,36,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,14,0,0,0,0,31,4,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,37,22,0,0,0,0,31,4,0,0,0,0,0,0,0,32,0,0,0,0,9,0] >;

C2.D8⋊D5 in GAP, Magma, Sage, TeX

C_2.D_8\rtimes D_5
% in TeX

G:=Group("C2.D8:D5");
// GroupNames label

G:=SmallGroup(320,512);
// by ID

G=gap.SmallGroup(320,512);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,268,851,102,12550]);
// Polycyclic

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

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