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G = C2.D87D5order 320 = 26·5

7th semidirect product of C2.D8 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.D87D5, C4⋊C4.53D10, (C2×C8).30D10, C20.43(C4○D4), C4.83(C4○D20), C10.30(C4○D8), C10.Q1622C2, C20.Q822C2, (C22×D5).38D4, C22.234(D4×D5), D101C8.11C2, C20.44D427C2, C2.15(D83D5), (C2×C40).244C22, (C2×C20).304C23, D102Q8.10C2, C4.31(Q82D5), (C2×Dic5).225D4, C55(C23.20D4), C2.24(Q16⋊D5), C10.72(C8.C22), C4⋊Dic5.127C22, (C2×Dic10).96C22, C2.18(D10.13D4), C10.48(C22.D4), (C5×C2.D8)⋊14C2, C4⋊C47D5.9C2, (C2×C4×D5).46C22, (C2×C10).309(C2×D4), (C5×C4⋊C4).97C22, (C2×C52C8).73C22, (C2×C4).407(C22×D5), SmallGroup(320,515)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C2.D87D5
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C2.D87D5
C5C10C2×C20 — C2.D87D5
C1C22C2×C4C2.D8

Generators and relations for C2.D87D5
 G = < a,b,c,d,e | a2=b8=d5=e2=1, c2=a, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, cd=dc, ece=ab4c, ede=d-1 >

Subgroups: 382 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C20.Q8, C10.Q16, C20.44D4, D101C8, C5×C2.D8, C4⋊C47D5, D102Q8, C2.D87D5
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, C23.20D4, C4○D20, D4×D5, Q82D5, D10.13D4, D83D5, Q16⋊D5, C2.D87D5

Smallest permutation representation of C2.D87D5
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 26)(2 126)(3 28)(4 128)(5 30)(6 122)(7 32)(8 124)(9 158)(10 147)(11 160)(12 149)(13 154)(14 151)(15 156)(16 145)(18 93)(20 95)(22 89)(24 91)(25 71)(27 65)(29 67)(31 69)(33 57)(34 86)(35 59)(36 88)(37 61)(38 82)(39 63)(40 84)(41 45)(42 98)(43 47)(44 100)(46 102)(48 104)(49 110)(50 136)(51 112)(52 130)(53 106)(54 132)(55 108)(56 134)(58 76)(60 78)(62 80)(64 74)(66 127)(68 121)(70 123)(72 125)(73 83)(75 85)(77 87)(79 81)(97 101)(99 103)(105 144)(107 138)(109 140)(111 142)(113 150)(114 155)(115 152)(116 157)(117 146)(118 159)(119 148)(120 153)(129 143)(131 137)(133 139)(135 141)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444455888810···102020202020···2040···40
size111120224481010202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8C4○D20C8.C22Q82D5D4×D5D83D5Q16⋊D5
kernelC2.D87D5C20.Q8C10.Q16C20.44D4D101C8C5×C2.D8C4⋊C47D5D102Q8C2×Dic5C22×D5C2.D8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111111124424812244

Matrix representation of C2.D87D5 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
233500
61800
0030
003414
,
22800
133900
001535
001026
,
6100
40000
0010
0001
,
0100
1000
0010
00540
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[23,6,0,0,35,18,0,0,0,0,3,34,0,0,0,14],[2,13,0,0,28,39,0,0,0,0,15,10,0,0,35,26],[6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,5,0,0,0,40] >;

C2.D87D5 in GAP, Magma, Sage, TeX

C_2.D_8\rtimes_7D_5
% in TeX

G:=Group("C2.D8:7D5");
// GroupNames label

G:=SmallGroup(320,515);
// by ID

G=gap.SmallGroup(320,515);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,268,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=d^5=e^2=1,c^2=a,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e=a*b^4*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽