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G = C5×C4⋊D8order 320 = 26·5

Direct product of C5 and C4⋊D8

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

Aliases: C5×C4⋊D8, C209D8, C42(C5×D8), C4⋊C83C10, D41(C5×D4), (C2×D8)⋊3C10, (C5×D4)⋊12D4, (C4×D4)⋊3C10, C2.5(C10×D8), (C10×D8)⋊17C2, (D4×C20)⋊32C2, C41D42C10, C10.77(C2×D8), C4.31(D4×C10), D4⋊C46C10, C20.392(C2×D4), (C2×C20).321D4, C42.14(C2×C10), C22.83(D4×C10), C20.341(C4○D4), (C2×C40).256C22, (C4×C20).256C22, (C2×C20).918C23, C10.142(C4⋊D4), C10.134(C8⋊C22), (D4×C10).185C22, (C5×C4⋊C8)⋊22C2, (C2×C8).3(C2×C10), C4.40(C5×C4○D4), C2.9(C5×C8⋊C22), (C5×C41D4)⋊12C2, C4⋊C4.51(C2×C10), (C2×C4).127(C5×D4), C2.11(C5×C4⋊D4), (C5×D4⋊C4)⋊29C2, (C2×D4).55(C2×C10), (C2×C10).639(C2×D4), (C5×C4⋊C4).372C22, (C2×C4).93(C22×C10), SmallGroup(320,960)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C4⋊D8
C1C2C4C2×C4C2×C20D4×C10C5×C41D4 — C5×C4⋊D8
C1C2C2×C4 — C5×C4⋊D8
C1C2×C10C4×C20 — C5×C4⋊D8

Generators and relations for C5×C4⋊D8
 G = < a,b,c,d | a5=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 314 in 140 conjugacy classes, 58 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4 [×9], C23 [×3], C10 [×3], C10 [×4], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×2], C2×D4 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×10], D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C40 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4 [×9], C22×C10 [×3], C4⋊D8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40 [×2], C5×D8 [×4], C22×C20, D4×C10, D4×C10 [×2], D4×C10 [×2], C5×D4⋊C4 [×2], C5×C4⋊C8, D4×C20, C5×C41D4, C10×D8 [×2], C5×C4⋊D8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], D8 [×2], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C2×D8, C8⋊C22, C5×D4 [×4], C22×C10, C4⋊D8, C5×D8 [×2], D4×C10 [×2], C5×C4○D4, C5×C4⋊D4, C10×D8, C5×C8⋊C22, C5×C4⋊D8

Smallest permutation representation of C5×C4⋊D8
On 160 points
Generators in S160
(1 41 89 56 81)(2 42 90 49 82)(3 43 91 50 83)(4 44 92 51 84)(5 45 93 52 85)(6 46 94 53 86)(7 47 95 54 87)(8 48 96 55 88)(9 115 139 17 131)(10 116 140 18 132)(11 117 141 19 133)(12 118 142 20 134)(13 119 143 21 135)(14 120 144 22 136)(15 113 137 23 129)(16 114 138 24 130)(25 124 155 33 147)(26 125 156 34 148)(27 126 157 35 149)(28 127 158 36 150)(29 128 159 37 151)(30 121 160 38 152)(31 122 153 39 145)(32 123 154 40 146)(57 74 107 66 99)(58 75 108 67 100)(59 76 109 68 101)(60 77 110 69 102)(61 78 111 70 103)(62 79 112 71 104)(63 80 105 72 97)(64 73 106 65 98)
(1 114 27 100)(2 101 28 115)(3 116 29 102)(4 103 30 117)(5 118 31 104)(6 97 32 119)(7 120 25 98)(8 99 26 113)(9 82 68 150)(10 151 69 83)(11 84 70 152)(12 145 71 85)(13 86 72 146)(14 147 65 87)(15 88 66 148)(16 149 67 81)(17 90 76 158)(18 159 77 91)(19 92 78 160)(20 153 79 93)(21 94 80 154)(22 155 73 95)(23 96 74 156)(24 157 75 89)(33 106 54 136)(34 129 55 107)(35 108 56 130)(36 131 49 109)(37 110 50 132)(38 133 51 111)(39 112 52 134)(40 135 53 105)(41 138 126 58)(42 59 127 139)(43 140 128 60)(44 61 121 141)(45 142 122 62)(46 63 123 143)(47 144 124 64)(48 57 125 137)
(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)
(1 100)(2 99)(3 98)(4 97)(5 104)(6 103)(7 102)(8 101)(9 148)(10 147)(11 146)(12 145)(13 152)(14 151)(15 150)(16 149)(17 156)(18 155)(19 154)(20 153)(21 160)(22 159)(23 158)(24 157)(25 116)(26 115)(27 114)(28 113)(29 120)(30 119)(31 118)(32 117)(33 132)(34 131)(35 130)(36 129)(37 136)(38 135)(39 134)(40 133)(41 58)(42 57)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 107)(50 106)(51 105)(52 112)(53 111)(54 110)(55 109)(56 108)(65 83)(66 82)(67 81)(68 88)(69 87)(70 86)(71 85)(72 84)(73 91)(74 90)(75 89)(76 96)(77 95)(78 94)(79 93)(80 92)(121 143)(122 142)(123 141)(124 140)(125 139)(126 138)(127 137)(128 144)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B5C5D8A8B8C8D10A···10L10M···10T10U···10AB20A···20P20Q···20AB40A···40P
order1222222244444445555888810···1010···1010···1020···2020···2040···40
size111144882222444111144441···14···48···82···24···44···4

95 irreducible representations

dim1111111111112222222244
type++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4D8C4○D4C5×D4C5×D4C5×D8C5×C4○D4C8⋊C22C5×C8⋊C22
kernelC5×C4⋊D8C5×D4⋊C4C5×C4⋊C8D4×C20C5×C41D4C10×D8C4⋊D8D4⋊C4C4⋊C8C4×D4C41D4C2×D8C2×C20C5×D4C20C20C2×C4D4C4C4C10C2
# reps12111248444822428816814

Matrix representation of C5×C4⋊D8 in GL4(𝔽41) generated by

1000
0100
00180
00018
,
403900
1100
00400
00040
,
403900
0100
00024
002924
,
403900
0100
00400
00401
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,39,1,0,0,0,0,0,29,0,0,24,24],[40,0,0,0,39,1,0,0,0,0,40,40,0,0,0,1] >;

C5×C4⋊D8 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes D_8
% in TeX

G:=Group("C5xC4:D8");
// GroupNames label

G:=SmallGroup(320,960);
// by ID

G=gap.SmallGroup(320,960);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽