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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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, 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)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 92)(18 93)(19 94)(20 95)(21 96)(22 89)(23 90)(24 91)(25 124)(26 125)(27 126)(28 127)(29 128)(30 121)(31 122)(32 123)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 101)(42 102)(43 103)(44 104)(45 97)(46 98)(47 99)(48 100)(49 141)(50 142)(51 143)(52 144)(53 137)(54 138)(55 139)(56 140)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 129)(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)(9 127 117 28)(10 126 118 27)(11 125 119 26)(12 124 120 25)(13 123 113 32)(14 122 114 31)(15 121 115 30)(16 128 116 29)(17 44 92 104)(18 43 93 103)(19 42 94 102)(20 41 95 101)(21 48 96 100)(22 47 89 99)(23 46 90 98)(24 45 91 97)(33 133 75 108)(34 132 76 107)(35 131 77 106)(36 130 78 105)(37 129 79 112)(38 136 80 111)(39 135 73 110)(40 134 74 109)(49 87 141 59)(50 86 142 58)(51 85 143 57)(52 84 144 64)(53 83 137 63)(54 82 138 62)(55 81 139 61)(56 88 140 60)
(1 61 17 37 26)(2 62 18 38 27)(3 63 19 39 28)(4 64 20 40 29)(5 57 21 33 30)(6 58 22 34 31)(7 59 23 35 32)(8 60 24 36 25)(9 150 53 42 135)(10 151 54 43 136)(11 152 55 44 129)(12 145 56 45 130)(13 146 49 46 131)(14 147 50 47 132)(15 148 51 48 133)(16 149 52 41 134)(65 82 93 80 126)(66 83 94 73 127)(67 84 95 74 128)(68 85 96 75 121)(69 86 89 76 122)(70 87 90 77 123)(71 88 91 78 124)(72 81 92 79 125)(97 105 120 157 140)(98 106 113 158 141)(99 107 114 159 142)(100 108 115 160 143)(101 109 116 153 144)(102 110 117 154 137)(103 111 118 155 138)(104 112 119 156 139)
(1 113)(2 10)(3 115)(4 12)(5 117)(6 14)(7 119)(8 16)(9 68)(11 70)(13 72)(15 66)(17 98)(18 43)(19 100)(20 45)(21 102)(22 47)(23 104)(24 41)(25 149)(26 158)(27 151)(28 160)(29 145)(30 154)(31 147)(32 156)(33 137)(34 50)(35 139)(36 52)(37 141)(38 54)(39 143)(40 56)(42 96)(44 90)(46 92)(48 94)(49 79)(51 73)(53 75)(55 77)(57 110)(58 132)(59 112)(60 134)(61 106)(62 136)(63 108)(64 130)(65 118)(67 120)(69 114)(71 116)(74 140)(76 142)(78 144)(80 138)(81 131)(82 111)(83 133)(84 105)(85 135)(86 107)(87 129)(88 109)(89 99)(91 101)(93 103)(95 97)(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)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,124)(26,125)(27,126)(28,127)(29,128)(30,121)(31,122)(32,123)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,129)(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)(9,127,117,28)(10,126,118,27)(11,125,119,26)(12,124,120,25)(13,123,113,32)(14,122,114,31)(15,121,115,30)(16,128,116,29)(17,44,92,104)(18,43,93,103)(19,42,94,102)(20,41,95,101)(21,48,96,100)(22,47,89,99)(23,46,90,98)(24,45,91,97)(33,133,75,108)(34,132,76,107)(35,131,77,106)(36,130,78,105)(37,129,79,112)(38,136,80,111)(39,135,73,110)(40,134,74,109)(49,87,141,59)(50,86,142,58)(51,85,143,57)(52,84,144,64)(53,83,137,63)(54,82,138,62)(55,81,139,61)(56,88,140,60), (1,61,17,37,26)(2,62,18,38,27)(3,63,19,39,28)(4,64,20,40,29)(5,57,21,33,30)(6,58,22,34,31)(7,59,23,35,32)(8,60,24,36,25)(9,150,53,42,135)(10,151,54,43,136)(11,152,55,44,129)(12,145,56,45,130)(13,146,49,46,131)(14,147,50,47,132)(15,148,51,48,133)(16,149,52,41,134)(65,82,93,80,126)(66,83,94,73,127)(67,84,95,74,128)(68,85,96,75,121)(69,86,89,76,122)(70,87,90,77,123)(71,88,91,78,124)(72,81,92,79,125)(97,105,120,157,140)(98,106,113,158,141)(99,107,114,159,142)(100,108,115,160,143)(101,109,116,153,144)(102,110,117,154,137)(103,111,118,155,138)(104,112,119,156,139), (1,113)(2,10)(3,115)(4,12)(5,117)(6,14)(7,119)(8,16)(9,68)(11,70)(13,72)(15,66)(17,98)(18,43)(19,100)(20,45)(21,102)(22,47)(23,104)(24,41)(25,149)(26,158)(27,151)(28,160)(29,145)(30,154)(31,147)(32,156)(33,137)(34,50)(35,139)(36,52)(37,141)(38,54)(39,143)(40,56)(42,96)(44,90)(46,92)(48,94)(49,79)(51,73)(53,75)(55,77)(57,110)(58,132)(59,112)(60,134)(61,106)(62,136)(63,108)(64,130)(65,118)(67,120)(69,114)(71,116)(74,140)(76,142)(78,144)(80,138)(81,131)(82,111)(83,133)(84,105)(85,135)(86,107)(87,129)(88,109)(89,99)(91,101)(93,103)(95,97)(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)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,124)(26,125)(27,126)(28,127)(29,128)(30,121)(31,122)(32,123)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,129)(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)(9,127,117,28)(10,126,118,27)(11,125,119,26)(12,124,120,25)(13,123,113,32)(14,122,114,31)(15,121,115,30)(16,128,116,29)(17,44,92,104)(18,43,93,103)(19,42,94,102)(20,41,95,101)(21,48,96,100)(22,47,89,99)(23,46,90,98)(24,45,91,97)(33,133,75,108)(34,132,76,107)(35,131,77,106)(36,130,78,105)(37,129,79,112)(38,136,80,111)(39,135,73,110)(40,134,74,109)(49,87,141,59)(50,86,142,58)(51,85,143,57)(52,84,144,64)(53,83,137,63)(54,82,138,62)(55,81,139,61)(56,88,140,60), (1,61,17,37,26)(2,62,18,38,27)(3,63,19,39,28)(4,64,20,40,29)(5,57,21,33,30)(6,58,22,34,31)(7,59,23,35,32)(8,60,24,36,25)(9,150,53,42,135)(10,151,54,43,136)(11,152,55,44,129)(12,145,56,45,130)(13,146,49,46,131)(14,147,50,47,132)(15,148,51,48,133)(16,149,52,41,134)(65,82,93,80,126)(66,83,94,73,127)(67,84,95,74,128)(68,85,96,75,121)(69,86,89,76,122)(70,87,90,77,123)(71,88,91,78,124)(72,81,92,79,125)(97,105,120,157,140)(98,106,113,158,141)(99,107,114,159,142)(100,108,115,160,143)(101,109,116,153,144)(102,110,117,154,137)(103,111,118,155,138)(104,112,119,156,139), (1,113)(2,10)(3,115)(4,12)(5,117)(6,14)(7,119)(8,16)(9,68)(11,70)(13,72)(15,66)(17,98)(18,43)(19,100)(20,45)(21,102)(22,47)(23,104)(24,41)(25,149)(26,158)(27,151)(28,160)(29,145)(30,154)(31,147)(32,156)(33,137)(34,50)(35,139)(36,52)(37,141)(38,54)(39,143)(40,56)(42,96)(44,90)(46,92)(48,94)(49,79)(51,73)(53,75)(55,77)(57,110)(58,132)(59,112)(60,134)(61,106)(62,136)(63,108)(64,130)(65,118)(67,120)(69,114)(71,116)(74,140)(76,142)(78,144)(80,138)(81,131)(82,111)(83,133)(84,105)(85,135)(86,107)(87,129)(88,109)(89,99)(91,101)(93,103)(95,97)(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),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,92),(18,93),(19,94),(20,95),(21,96),(22,89),(23,90),(24,91),(25,124),(26,125),(27,126),(28,127),(29,128),(30,121),(31,122),(32,123),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,101),(42,102),(43,103),(44,104),(45,97),(46,98),(47,99),(48,100),(49,141),(50,142),(51,143),(52,144),(53,137),(54,138),(55,139),(56,140),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,129),(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),(9,127,117,28),(10,126,118,27),(11,125,119,26),(12,124,120,25),(13,123,113,32),(14,122,114,31),(15,121,115,30),(16,128,116,29),(17,44,92,104),(18,43,93,103),(19,42,94,102),(20,41,95,101),(21,48,96,100),(22,47,89,99),(23,46,90,98),(24,45,91,97),(33,133,75,108),(34,132,76,107),(35,131,77,106),(36,130,78,105),(37,129,79,112),(38,136,80,111),(39,135,73,110),(40,134,74,109),(49,87,141,59),(50,86,142,58),(51,85,143,57),(52,84,144,64),(53,83,137,63),(54,82,138,62),(55,81,139,61),(56,88,140,60)], [(1,61,17,37,26),(2,62,18,38,27),(3,63,19,39,28),(4,64,20,40,29),(5,57,21,33,30),(6,58,22,34,31),(7,59,23,35,32),(8,60,24,36,25),(9,150,53,42,135),(10,151,54,43,136),(11,152,55,44,129),(12,145,56,45,130),(13,146,49,46,131),(14,147,50,47,132),(15,148,51,48,133),(16,149,52,41,134),(65,82,93,80,126),(66,83,94,73,127),(67,84,95,74,128),(68,85,96,75,121),(69,86,89,76,122),(70,87,90,77,123),(71,88,91,78,124),(72,81,92,79,125),(97,105,120,157,140),(98,106,113,158,141),(99,107,114,159,142),(100,108,115,160,143),(101,109,116,153,144),(102,110,117,154,137),(103,111,118,155,138),(104,112,119,156,139)], [(1,113),(2,10),(3,115),(4,12),(5,117),(6,14),(7,119),(8,16),(9,68),(11,70),(13,72),(15,66),(17,98),(18,43),(19,100),(20,45),(21,102),(22,47),(23,104),(24,41),(25,149),(26,158),(27,151),(28,160),(29,145),(30,154),(31,147),(32,156),(33,137),(34,50),(35,139),(36,52),(37,141),(38,54),(39,143),(40,56),(42,96),(44,90),(46,92),(48,94),(49,79),(51,73),(53,75),(55,77),(57,110),(58,132),(59,112),(60,134),(61,106),(62,136),(63,108),(64,130),(65,118),(67,120),(69,114),(71,116),(74,140),(76,142),(78,144),(80,138),(81,131),(82,111),(83,133),(84,105),(85,135),(86,107),(87,129),(88,109),(89,99),(91,101),(93,103),(95,97),(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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