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G = C5×C86D4order 320 = 26·5

Direct product of C5 and C86D4

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

Aliases: C5×C86D4, C4028D4, C2012M4(2), C86(C5×D4), C4⋊C816C10, (C4×C40)⋊31C2, (C4×C8)⋊15C10, C4⋊C4.8C20, (C2×D4).9C20, (C4×D4).3C10, C4.81(D4×C10), C2.11(D4×C20), C41(C5×M4(2)), C22⋊C814C10, (D4×C20).18C2, (D4×C10).34C4, C20.486(C2×D4), C10.143(C4×D4), C22⋊C4.5C20, C10.74(C8○D4), C42.69(C2×C10), C23.12(C2×C20), C20.355(C4○D4), (C2×M4(2))⋊15C10, (C10×M4(2))⋊33C2, (C4×C20).354C22, (C2×C20).992C23, (C2×C40).328C22, C2.10(C10×M4(2)), C10.88(C2×M4(2)), C22.48(C22×C20), (C22×C20).418C22, (C5×C4⋊C8)⋊35C2, C2.8(C5×C8○D4), (C5×C4⋊C4).33C4, C4.53(C5×C4○D4), (C5×C22⋊C8)⋊31C2, (C2×C8).53(C2×C10), (C2×C4).29(C2×C20), (C2×C20).386(C2×C4), (C5×C22⋊C4).20C4, (C22×C4).36(C2×C10), (C22×C10).92(C2×C4), (C2×C4).160(C22×C10), (C2×C10).343(C22×C4), SmallGroup(320,937)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C86D4
C1C2C4C2×C4C2×C20C2×C40C5×C22⋊C8 — C5×C86D4
C1C22 — C5×C86D4
C1C2×C20 — C5×C86D4

Generators and relations for C5×C86D4
 G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C40 [×2], C40 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C86D4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×C20 [×2], D4×C10, C4×C40, C5×C22⋊C8 [×2], C5×C4⋊C8, D4×C20, C10×M4(2) [×2], C5×C86D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], M4(2) [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×M4(2), C8○D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C86D4, C5×M4(2) [×2], C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×M4(2), C5×C8○D4, C5×C86D4

Smallest permutation representation of C5×C86D4
On 160 points
Generators in S160
(1 51 14 57 17)(2 52 15 58 18)(3 53 16 59 19)(4 54 9 60 20)(5 55 10 61 21)(6 56 11 62 22)(7 49 12 63 23)(8 50 13 64 24)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 157 116 149 108)(42 158 117 150 109)(43 159 118 151 110)(44 160 119 152 111)(45 153 120 145 112)(46 154 113 146 105)(47 155 114 147 106)(48 156 115 148 107)(89 122 142 101 129)(90 123 143 102 130)(91 124 144 103 131)(92 125 137 104 132)(93 126 138 97 133)(94 127 139 98 134)(95 128 140 99 135)(96 121 141 100 136)
(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)
(1 105 27 101)(2 106 28 102)(3 107 29 103)(4 108 30 104)(5 109 31 97)(6 110 32 98)(7 111 25 99)(8 112 26 100)(9 157 78 92)(10 158 79 93)(11 159 80 94)(12 160 73 95)(13 153 74 96)(14 154 75 89)(15 155 76 90)(16 156 77 91)(17 146 67 142)(18 147 68 143)(19 148 69 144)(20 149 70 137)(21 150 71 138)(22 151 72 139)(23 152 65 140)(24 145 66 141)(33 128 63 119)(34 121 64 120)(35 122 57 113)(36 123 58 114)(37 124 59 115)(38 125 60 116)(39 126 61 117)(40 127 62 118)(41 85 132 54)(42 86 133 55)(43 87 134 56)(44 88 135 49)(45 81 136 50)(46 82 129 51)(47 83 130 52)(48 84 131 53)
(1 101)(2 98)(3 103)(4 100)(5 97)(6 102)(7 99)(8 104)(9 96)(10 93)(11 90)(12 95)(13 92)(14 89)(15 94)(16 91)(17 142)(18 139)(19 144)(20 141)(21 138)(22 143)(23 140)(24 137)(25 111)(26 108)(27 105)(28 110)(29 107)(30 112)(31 109)(32 106)(33 119)(34 116)(35 113)(36 118)(37 115)(38 120)(39 117)(40 114)(41 81)(42 86)(43 83)(44 88)(45 85)(46 82)(47 87)(48 84)(49 135)(50 132)(51 129)(52 134)(53 131)(54 136)(55 133)(56 130)(57 122)(58 127)(59 124)(60 121)(61 126)(62 123)(63 128)(64 125)(65 152)(66 149)(67 146)(68 151)(69 148)(70 145)(71 150)(72 147)(73 160)(74 157)(75 154)(76 159)(77 156)(78 153)(79 158)(80 155)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B5C5D8A···8H8I8J8K8L10A···10L10M···10T20A···20P20Q···20AF20AG···20AN40A···40AF40AG···40AV
order122222444444444455558···8888810···1010···1020···2020···2020···2040···4040···40
size111144111122224411112···244441···14···41···12···24···42···24···4

140 irreducible representations

dim11111111111111111122222222
type+++++++
imageC1C2C2C2C2C2C4C4C4C5C10C10C10C10C10C20C20C20D4M4(2)C4○D4C8○D4C5×D4C5×M4(2)C5×C4○D4C5×C8○D4
kernelC5×C86D4C4×C40C5×C22⋊C8C5×C4⋊C8D4×C20C10×M4(2)C5×C22⋊C4C5×C4⋊C4D4×C10C86D4C4×C8C22⋊C8C4⋊C8C4×D4C2×M4(2)C22⋊C4C4⋊C4C2×D4C40C20C20C10C8C4C4C2
# reps11211242244844816882424816816

Matrix representation of C5×C86D4 in GL4(𝔽41) generated by

18000
01800
0010
0001
,
163900
212500
0001
0090
,
34600
19700
0010
0001
,
34600
33700
0010
00040
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,21,0,0,39,25,0,0,0,0,0,9,0,0,1,0],[34,19,0,0,6,7,0,0,0,0,1,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;

C5×C86D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_6D_4
% in TeX

G:=Group("C5xC8:6D4");
// GroupNames label

G:=SmallGroup(320,937);
// by ID

G=gap.SmallGroup(320,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1731,436,124]);
// Polycyclic

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

׿
×
𝔽