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G = C5×D4⋊C8order 320 = 26·5

Direct product of C5 and D4⋊C8

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

Aliases: C5×D4⋊C8, D4⋊C40, C20.67D8, C20.54SD16, C20.42M4(2), C4⋊C81C10, (C4×C8)⋊1C10, (C4×C40)⋊2C2, (C5×D4)⋊5C8, C4⋊C4.3C20, C4.1(C2×C40), C4.16(C5×D8), C10.32C4≀C2, C20.64(C2×C8), (C2×D4).4C20, (C4×D4).1C10, (D4×C10).27C4, (D4×C20).16C2, (C2×C20).528D4, C4.13(C5×SD16), C4.1(C5×M4(2)), C42.62(C2×C10), C10.37(C22⋊C8), (C4×C20).346C22, C10.47(D4⋊C4), (C5×C4⋊C8)⋊3C2, C2.1(C5×C4≀C2), (C5×C4⋊C4).28C4, (C2×C4).93(C5×D4), C2.5(C5×C22⋊C8), (C2×C4).38(C2×C20), C2.1(C5×D4⋊C4), (C2×C20).431(C2×C4), C22.25(C5×C22⋊C4), (C2×C10).184(C22⋊C4), SmallGroup(320,130)

Series: Derived Chief Lower central Upper central

C1C4 — C5×D4⋊C8
C1C2C22C2×C4C42C4×C20C5×C4⋊C8 — C5×D4⋊C8
C1C2C4 — C5×D4⋊C8
C1C2×C20C4×C20 — C5×D4⋊C8

Generators and relations for C5×D4⋊C8
 G = < a,b,c,d | a5=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 154 in 82 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, C20 [×4], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, C4⋊C8, C4×D4, C40 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4⋊C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40 [×2], C22×C20, D4×C10, C4×C40, C5×C4⋊C8, D4×C20, C5×D4⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), D8, SD16, C20 [×2], C2×C10, C22⋊C8, D4⋊C4, C4≀C2, C40 [×2], C2×C20, C5×D4 [×2], D4⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×D8, C5×SD16, C5×C22⋊C8, C5×D4⋊C4, C5×C4≀C2, C5×D4⋊C8

Smallest permutation representation of C5×D4⋊C8
On 160 points
Generators in S160
(1 12 113 105 97)(2 13 114 106 98)(3 14 115 107 99)(4 15 116 108 100)(5 16 117 109 101)(6 9 118 110 102)(7 10 119 111 103)(8 11 120 112 104)(17 93 41 33 25)(18 94 42 34 26)(19 95 43 35 27)(20 96 44 36 28)(21 89 45 37 29)(22 90 46 38 30)(23 91 47 39 31)(24 92 48 40 32)(49 87 73 65 57)(50 88 74 66 58)(51 81 75 67 59)(52 82 76 68 60)(53 83 77 69 61)(54 84 78 70 62)(55 85 79 71 63)(56 86 80 72 64)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)
(1 55 127 17)(2 18 128 56)(3 49 121 19)(4 20 122 50)(5 51 123 21)(6 22 124 52)(7 53 125 23)(8 24 126 54)(9 90 156 82)(10 83 157 91)(11 92 158 84)(12 85 159 93)(13 94 160 86)(14 87 153 95)(15 96 154 88)(16 81 155 89)(25 97 63 135)(26 136 64 98)(27 99 57 129)(28 130 58 100)(29 101 59 131)(30 132 60 102)(31 103 61 133)(32 134 62 104)(33 105 71 143)(34 144 72 106)(35 107 65 137)(36 138 66 108)(37 109 67 139)(38 140 68 110)(39 111 69 141)(40 142 70 112)(41 113 79 151)(42 152 80 114)(43 115 73 145)(44 146 74 116)(45 117 75 147)(46 148 76 118)(47 119 77 149)(48 150 78 120)
(1 17)(2 128)(3 19)(4 122)(5 21)(6 124)(7 23)(8 126)(9 156)(10 91)(11 158)(12 93)(13 160)(14 95)(15 154)(16 89)(25 97)(27 99)(29 101)(31 103)(33 105)(35 107)(37 109)(39 111)(41 113)(43 115)(45 117)(47 119)(49 121)(51 123)(53 125)(55 127)(57 129)(59 131)(61 133)(63 135)(65 137)(67 139)(69 141)(71 143)(73 145)(75 147)(77 149)(79 151)(81 155)(83 157)(85 159)(87 153)(98 136)(100 130)(102 132)(104 134)(106 144)(108 138)(110 140)(112 142)(114 152)(116 146)(118 148)(120 150)
(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)

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

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

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

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

dim111111111111112222222222
type++++++
imageC1C2C2C2C4C4C5C8C10C10C10C20C20C40D4M4(2)D8SD16C4≀C2C5×D4C5×M4(2)C5×D8C5×SD16C5×C4≀C2
kernelC5×D4⋊C8C4×C40C5×C4⋊C8D4×C20C5×C4⋊C4D4×C10D4⋊C8C5×D4C4×C8C4⋊C8C4×D4C4⋊C4C2×D4D4C2×C20C20C20C20C10C2×C4C4C4C4C2
# reps11112248444883222224888816

Matrix representation of C5×D4⋊C8 in GL3(𝔽41) generated by

100
0100
0010
,
100
0139
0140
,
4000
0139
0040
,
2700
008
040
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[1,0,0,0,1,1,0,39,40],[40,0,0,0,1,0,0,39,40],[27,0,0,0,0,4,0,8,0] >;

C5×D4⋊C8 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes C_8
% in TeX

G:=Group("C5xD4:C8");
// GroupNames label

G:=SmallGroup(320,130);
// by ID

G=gap.SmallGroup(320,130);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,1410,136,172]);
// Polycyclic

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

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