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## G = C4×D5⋊C8order 320 = 26·5

### Direct product of C4 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C4×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C4×D5⋊C8
 Lower central C5 — C4×D5⋊C8
 Upper central C1 — C42

Generators and relations for C4×D5⋊C8
G = < a,b,c,d | a4=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 426 in 162 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×2], C2×C4 [×15], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×12], C22×C4 [×3], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C4×C8 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×8], C4×D5 [×12], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C4×C8, C4×Dic5, C4×Dic5 [×2], C4×C20, D5⋊C8 [×8], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C4×C5⋊C8 [×4], D5×C42, C2×D5⋊C8 [×2], C4×D5⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], F5, C4×C8 [×4], C2×C42, C22×C8 [×2], C2×F5 [×3], C2×C4×C8, D5⋊C8 [×4], C4×F5 [×2], C22×F5, C2×D5⋊C8 [×2], C2×C4×F5, C4×D5⋊C8

Smallest permutation representation of C4×D5⋊C8
On 160 points
Generators in S160
(1 17 103 75)(2 18 104 76)(3 19 97 77)(4 20 98 78)(5 21 99 79)(6 22 100 80)(7 23 101 73)(8 24 102 74)(9 65 55 133)(10 66 56 134)(11 67 49 135)(12 68 50 136)(13 69 51 129)(14 70 52 130)(15 71 53 131)(16 72 54 132)(25 90 152 158)(26 91 145 159)(27 92 146 160)(28 93 147 153)(29 94 148 154)(30 95 149 155)(31 96 150 156)(32 89 151 157)(33 122 109 83)(34 123 110 84)(35 124 111 85)(36 125 112 86)(37 126 105 87)(38 127 106 88)(39 128 107 81)(40 121 108 82)(41 139 118 62)(42 140 119 63)(43 141 120 64)(44 142 113 57)(45 143 114 58)(46 144 115 59)(47 137 116 60)(48 138 117 61)
(1 66 61 39 26)(2 40 67 27 62)(3 28 33 63 68)(4 64 29 69 34)(5 70 57 35 30)(6 36 71 31 58)(7 32 37 59 72)(8 60 25 65 38)(9 88 74 116 158)(10 117 81 159 75)(11 160 118 76 82)(12 77 153 83 119)(13 84 78 120 154)(14 113 85 155 79)(15 156 114 80 86)(16 73 157 87 115)(17 56 48 128 91)(18 121 49 92 41)(19 93 122 42 50)(20 43 94 51 123)(21 52 44 124 95)(22 125 53 96 45)(23 89 126 46 54)(24 47 90 55 127)(97 147 109 140 136)(98 141 148 129 110)(99 130 142 111 149)(100 112 131 150 143)(101 151 105 144 132)(102 137 152 133 106)(103 134 138 107 145)(104 108 135 146 139)
(1 26)(2 62)(3 68)(4 34)(5 30)(6 58)(7 72)(8 38)(9 116)(10 81)(12 77)(13 120)(14 85)(16 73)(17 91)(18 41)(19 50)(20 123)(21 95)(22 45)(23 54)(24 127)(27 40)(28 63)(31 36)(32 59)(35 70)(39 66)(42 93)(43 51)(46 89)(47 55)(52 124)(56 128)(60 65)(64 69)(74 88)(75 159)(76 118)(78 84)(79 155)(80 114)(82 160)(86 156)(92 121)(96 125)(97 136)(98 110)(99 149)(100 143)(101 132)(102 106)(103 145)(104 139)(107 134)(108 146)(111 130)(112 150)(115 157)(119 153)(129 141)(133 137)(140 147)(144 151)
(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)| (1,17,103,75)(2,18,104,76)(3,19,97,77)(4,20,98,78)(5,21,99,79)(6,22,100,80)(7,23,101,73)(8,24,102,74)(9,65,55,133)(10,66,56,134)(11,67,49,135)(12,68,50,136)(13,69,51,129)(14,70,52,130)(15,71,53,131)(16,72,54,132)(25,90,152,158)(26,91,145,159)(27,92,146,160)(28,93,147,153)(29,94,148,154)(30,95,149,155)(31,96,150,156)(32,89,151,157)(33,122,109,83)(34,123,110,84)(35,124,111,85)(36,125,112,86)(37,126,105,87)(38,127,106,88)(39,128,107,81)(40,121,108,82)(41,139,118,62)(42,140,119,63)(43,141,120,64)(44,142,113,57)(45,143,114,58)(46,144,115,59)(47,137,116,60)(48,138,117,61), (1,66,61,39,26)(2,40,67,27,62)(3,28,33,63,68)(4,64,29,69,34)(5,70,57,35,30)(6,36,71,31,58)(7,32,37,59,72)(8,60,25,65,38)(9,88,74,116,158)(10,117,81,159,75)(11,160,118,76,82)(12,77,153,83,119)(13,84,78,120,154)(14,113,85,155,79)(15,156,114,80,86)(16,73,157,87,115)(17,56,48,128,91)(18,121,49,92,41)(19,93,122,42,50)(20,43,94,51,123)(21,52,44,124,95)(22,125,53,96,45)(23,89,126,46,54)(24,47,90,55,127)(97,147,109,140,136)(98,141,148,129,110)(99,130,142,111,149)(100,112,131,150,143)(101,151,105,144,132)(102,137,152,133,106)(103,134,138,107,145)(104,108,135,146,139), (1,26)(2,62)(3,68)(4,34)(5,30)(6,58)(7,72)(8,38)(9,116)(10,81)(12,77)(13,120)(14,85)(16,73)(17,91)(18,41)(19,50)(20,123)(21,95)(22,45)(23,54)(24,127)(27,40)(28,63)(31,36)(32,59)(35,70)(39,66)(42,93)(43,51)(46,89)(47,55)(52,124)(56,128)(60,65)(64,69)(74,88)(75,159)(76,118)(78,84)(79,155)(80,114)(82,160)(86,156)(92,121)(96,125)(97,136)(98,110)(99,149)(100,143)(101,132)(102,106)(103,145)(104,139)(107,134)(108,146)(111,130)(112,150)(115,157)(119,153)(129,141)(133,137)(140,147)(144,151), (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( (1,17,103,75)(2,18,104,76)(3,19,97,77)(4,20,98,78)(5,21,99,79)(6,22,100,80)(7,23,101,73)(8,24,102,74)(9,65,55,133)(10,66,56,134)(11,67,49,135)(12,68,50,136)(13,69,51,129)(14,70,52,130)(15,71,53,131)(16,72,54,132)(25,90,152,158)(26,91,145,159)(27,92,146,160)(28,93,147,153)(29,94,148,154)(30,95,149,155)(31,96,150,156)(32,89,151,157)(33,122,109,83)(34,123,110,84)(35,124,111,85)(36,125,112,86)(37,126,105,87)(38,127,106,88)(39,128,107,81)(40,121,108,82)(41,139,118,62)(42,140,119,63)(43,141,120,64)(44,142,113,57)(45,143,114,58)(46,144,115,59)(47,137,116,60)(48,138,117,61), (1,66,61,39,26)(2,40,67,27,62)(3,28,33,63,68)(4,64,29,69,34)(5,70,57,35,30)(6,36,71,31,58)(7,32,37,59,72)(8,60,25,65,38)(9,88,74,116,158)(10,117,81,159,75)(11,160,118,76,82)(12,77,153,83,119)(13,84,78,120,154)(14,113,85,155,79)(15,156,114,80,86)(16,73,157,87,115)(17,56,48,128,91)(18,121,49,92,41)(19,93,122,42,50)(20,43,94,51,123)(21,52,44,124,95)(22,125,53,96,45)(23,89,126,46,54)(24,47,90,55,127)(97,147,109,140,136)(98,141,148,129,110)(99,130,142,111,149)(100,112,131,150,143)(101,151,105,144,132)(102,137,152,133,106)(103,134,138,107,145)(104,108,135,146,139), (1,26)(2,62)(3,68)(4,34)(5,30)(6,58)(7,72)(8,38)(9,116)(10,81)(12,77)(13,120)(14,85)(16,73)(17,91)(18,41)(19,50)(20,123)(21,95)(22,45)(23,54)(24,127)(27,40)(28,63)(31,36)(32,59)(35,70)(39,66)(42,93)(43,51)(46,89)(47,55)(52,124)(56,128)(60,65)(64,69)(74,88)(75,159)(76,118)(78,84)(79,155)(80,114)(82,160)(86,156)(92,121)(96,125)(97,136)(98,110)(99,149)(100,143)(101,132)(102,106)(103,145)(104,139)(107,134)(108,146)(111,130)(112,150)(115,157)(119,153)(129,141)(133,137)(140,147)(144,151), (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([(1,17,103,75),(2,18,104,76),(3,19,97,77),(4,20,98,78),(5,21,99,79),(6,22,100,80),(7,23,101,73),(8,24,102,74),(9,65,55,133),(10,66,56,134),(11,67,49,135),(12,68,50,136),(13,69,51,129),(14,70,52,130),(15,71,53,131),(16,72,54,132),(25,90,152,158),(26,91,145,159),(27,92,146,160),(28,93,147,153),(29,94,148,154),(30,95,149,155),(31,96,150,156),(32,89,151,157),(33,122,109,83),(34,123,110,84),(35,124,111,85),(36,125,112,86),(37,126,105,87),(38,127,106,88),(39,128,107,81),(40,121,108,82),(41,139,118,62),(42,140,119,63),(43,141,120,64),(44,142,113,57),(45,143,114,58),(46,144,115,59),(47,137,116,60),(48,138,117,61)], [(1,66,61,39,26),(2,40,67,27,62),(3,28,33,63,68),(4,64,29,69,34),(5,70,57,35,30),(6,36,71,31,58),(7,32,37,59,72),(8,60,25,65,38),(9,88,74,116,158),(10,117,81,159,75),(11,160,118,76,82),(12,77,153,83,119),(13,84,78,120,154),(14,113,85,155,79),(15,156,114,80,86),(16,73,157,87,115),(17,56,48,128,91),(18,121,49,92,41),(19,93,122,42,50),(20,43,94,51,123),(21,52,44,124,95),(22,125,53,96,45),(23,89,126,46,54),(24,47,90,55,127),(97,147,109,140,136),(98,141,148,129,110),(99,130,142,111,149),(100,112,131,150,143),(101,151,105,144,132),(102,137,152,133,106),(103,134,138,107,145),(104,108,135,146,139)], [(1,26),(2,62),(3,68),(4,34),(5,30),(6,58),(7,72),(8,38),(9,116),(10,81),(12,77),(13,120),(14,85),(16,73),(17,91),(18,41),(19,50),(20,123),(21,95),(22,45),(23,54),(24,127),(27,40),(28,63),(31,36),(32,59),(35,70),(39,66),(42,93),(43,51),(46,89),(47,55),(52,124),(56,128),(60,65),(64,69),(74,88),(75,159),(76,118),(78,84),(79,155),(80,114),(82,160),(86,156),(92,121),(96,125),(97,136),(98,110),(99,149),(100,143),(101,132),(102,106),(103,145),(104,139),(107,134),(108,146),(111,130),(112,150),(115,157),(119,153),(129,141),(133,137),(140,147),(144,151)], [(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)])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 5 8A ··· 8AF 10A 10B 10C 20A ··· 20L order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 8 ··· 8 10 10 10 20 ··· 20 size 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 4 5 ··· 5 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 F5 C2×F5 D5⋊C8 C4×F5 kernel C4×D5⋊C8 C4×C5⋊C8 D5×C42 C2×D5⋊C8 C4×Dic5 C4×C20 D5⋊C8 C2×C4×D5 C4×D5 C42 C2×C4 C4 C4 # reps 1 4 1 2 2 2 16 4 32 1 3 8 4

Matrix representation of C4×D5⋊C8 in GL5(𝔽41)

 1 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 40 0 0 0 1 34 0 0 0 0 0 40 7 0 0 0 34 7
,
 40 0 0 0 0 0 34 7 0 0 0 40 7 0 0 0 0 0 7 40 0 0 0 7 34
,
 27 0 0 0 0 0 0 0 9 0 0 0 0 0 9 0 32 22 0 0 0 0 9 0 0

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,40,34,0,0,0,0,0,40,34,0,0,0,7,7],[40,0,0,0,0,0,34,40,0,0,0,7,7,0,0,0,0,0,7,7,0,0,0,40,34],[27,0,0,0,0,0,0,0,32,0,0,0,0,22,9,0,9,0,0,0,0,0,9,0,0] >;

C4×D5⋊C8 in GAP, Magma, Sage, TeX

C_4\times D_5\rtimes C_8
% in TeX

G:=Group("C4xD5:C8");
// GroupNames label

G:=SmallGroup(320,1013);
// by ID

G=gap.SmallGroup(320,1013);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

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