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## G = C4×C5⋊2C8order 160 = 25·5

### Direct product of C4 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C4×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C4×C5⋊2C8
 Lower central C5 — C4×C5⋊2C8
 Upper central C1 — C42

Generators and relations for C4×C52C8
G = < a,b,c | a4=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×C52C8
Regular action on 160 points
Generators in S160
(1 85 61 69)(2 86 62 70)(3 87 63 71)(4 88 64 72)(5 81 57 65)(6 82 58 66)(7 83 59 67)(8 84 60 68)(9 91 20 136)(10 92 21 129)(11 93 22 130)(12 94 23 131)(13 95 24 132)(14 96 17 133)(15 89 18 134)(16 90 19 135)(25 73 121 149)(26 74 122 150)(27 75 123 151)(28 76 124 152)(29 77 125 145)(30 78 126 146)(31 79 127 147)(32 80 128 148)(33 45 100 141)(34 46 101 142)(35 47 102 143)(36 48 103 144)(37 41 104 137)(38 42 97 138)(39 43 98 139)(40 44 99 140)(49 116 110 158)(50 117 111 159)(51 118 112 160)(52 119 105 153)(53 120 106 154)(54 113 107 155)(55 114 108 156)(56 115 109 157)
(1 147 132 143 113)(2 114 144 133 148)(3 149 134 137 115)(4 116 138 135 150)(5 151 136 139 117)(6 118 140 129 152)(7 145 130 141 119)(8 120 142 131 146)(9 39 111 81 27)(10 28 82 112 40)(11 33 105 83 29)(12 30 84 106 34)(13 35 107 85 31)(14 32 86 108 36)(15 37 109 87 25)(16 26 88 110 38)(17 128 70 55 103)(18 104 56 71 121)(19 122 72 49 97)(20 98 50 65 123)(21 124 66 51 99)(22 100 52 67 125)(23 126 68 53 101)(24 102 54 69 127)(41 157 63 73 89)(42 90 74 64 158)(43 159 57 75 91)(44 92 76 58 160)(45 153 59 77 93)(46 94 78 60 154)(47 155 61 79 95)(48 96 80 62 156)
(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)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 5A 5B 8A ··· 8P 10A ··· 10F 20A ··· 20X order 1 2 2 2 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 1 ··· 1 2 2 5 ··· 5 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C8 D5 Dic5 D10 C5⋊2C8 C4×D5 kernel C4×C5⋊2C8 C2×C5⋊2C8 C4×C20 C5⋊2C8 C2×C20 C20 C42 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 16 2 4 2 16 8

Matrix representation of C4×C52C8 in GL3(𝔽41) generated by

 1 0 0 0 32 0 0 0 32
,
 1 0 0 0 6 40 0 1 0
,
 38 0 0 0 1 0 0 6 40
G:=sub<GL(3,GF(41))| [1,0,0,0,32,0,0,0,32],[1,0,0,0,6,1,0,40,0],[38,0,0,0,1,6,0,0,40] >;

C4×C52C8 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes_2C_8
% in TeX

G:=Group("C4xC5:2C8");
// GroupNames label

G:=SmallGroup(160,9);
// by ID

G=gap.SmallGroup(160,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,86,4613]);
// Polycyclic

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

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