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## G = C5×C4⋊C8order 160 = 25·5

### Direct product of C5 and C4⋊C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C4⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C40 — C5×C4⋊C8
 Lower central C1 — C2 — C5×C4⋊C8
 Upper central C1 — C2×C20 — C5×C4⋊C8

Generators and relations for C5×C4⋊C8
G = < a,b,c | a5=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×C4⋊C8
Regular action on 160 points
Generators in S160
(1 117 37 109 29)(2 118 38 110 30)(3 119 39 111 31)(4 120 40 112 32)(5 113 33 105 25)(6 114 34 106 26)(7 115 35 107 27)(8 116 36 108 28)(9 20 43 97 95)(10 21 44 98 96)(11 22 45 99 89)(12 23 46 100 90)(13 24 47 101 91)(14 17 48 102 92)(15 18 41 103 93)(16 19 42 104 94)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 67 127 41)(2 42 128 68)(3 69 121 43)(4 44 122 70)(5 71 123 45)(6 46 124 72)(7 65 125 47)(8 48 126 66)(9 111 77 129)(10 130 78 112)(11 105 79 131)(12 132 80 106)(13 107 73 133)(14 134 74 108)(15 109 75 135)(16 136 76 110)(17 54 146 28)(18 29 147 55)(19 56 148 30)(20 31 149 49)(21 50 150 32)(22 25 151 51)(23 52 152 26)(24 27 145 53)(33 159 59 89)(34 90 60 160)(35 153 61 91)(36 92 62 154)(37 155 63 93)(38 94 64 156)(39 157 57 95)(40 96 58 158)(81 137 97 119)(82 120 98 138)(83 139 99 113)(84 114 100 140)(85 141 101 115)(86 116 102 142)(87 143 103 117)(88 118 104 144)
(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)| (1,117,37,109,29)(2,118,38,110,30)(3,119,39,111,31)(4,120,40,112,32)(5,113,33,105,25)(6,114,34,106,26)(7,115,35,107,27)(8,116,36,108,28)(9,20,43,97,95)(10,21,44,98,96)(11,22,45,99,89)(12,23,46,100,90)(13,24,47,101,91)(14,17,48,102,92)(15,18,41,103,93)(16,19,42,104,94)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,67,127,41)(2,42,128,68)(3,69,121,43)(4,44,122,70)(5,71,123,45)(6,46,124,72)(7,65,125,47)(8,48,126,66)(9,111,77,129)(10,130,78,112)(11,105,79,131)(12,132,80,106)(13,107,73,133)(14,134,74,108)(15,109,75,135)(16,136,76,110)(17,54,146,28)(18,29,147,55)(19,56,148,30)(20,31,149,49)(21,50,150,32)(22,25,151,51)(23,52,152,26)(24,27,145,53)(33,159,59,89)(34,90,60,160)(35,153,61,91)(36,92,62,154)(37,155,63,93)(38,94,64,156)(39,157,57,95)(40,96,58,158)(81,137,97,119)(82,120,98,138)(83,139,99,113)(84,114,100,140)(85,141,101,115)(86,116,102,142)(87,143,103,117)(88,118,104,144), (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( (1,117,37,109,29)(2,118,38,110,30)(3,119,39,111,31)(4,120,40,112,32)(5,113,33,105,25)(6,114,34,106,26)(7,115,35,107,27)(8,116,36,108,28)(9,20,43,97,95)(10,21,44,98,96)(11,22,45,99,89)(12,23,46,100,90)(13,24,47,101,91)(14,17,48,102,92)(15,18,41,103,93)(16,19,42,104,94)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,67,127,41)(2,42,128,68)(3,69,121,43)(4,44,122,70)(5,71,123,45)(6,46,124,72)(7,65,125,47)(8,48,126,66)(9,111,77,129)(10,130,78,112)(11,105,79,131)(12,132,80,106)(13,107,73,133)(14,134,74,108)(15,109,75,135)(16,136,76,110)(17,54,146,28)(18,29,147,55)(19,56,148,30)(20,31,149,49)(21,50,150,32)(22,25,151,51)(23,52,152,26)(24,27,145,53)(33,159,59,89)(34,90,60,160)(35,153,61,91)(36,92,62,154)(37,155,63,93)(38,94,64,156)(39,157,57,95)(40,96,58,158)(81,137,97,119)(82,120,98,138)(83,139,99,113)(84,114,100,140)(85,141,101,115)(86,116,102,142)(87,143,103,117)(88,118,104,144), (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([[(1,117,37,109,29),(2,118,38,110,30),(3,119,39,111,31),(4,120,40,112,32),(5,113,33,105,25),(6,114,34,106,26),(7,115,35,107,27),(8,116,36,108,28),(9,20,43,97,95),(10,21,44,98,96),(11,22,45,99,89),(12,23,46,100,90),(13,24,47,101,91),(14,17,48,102,92),(15,18,41,103,93),(16,19,42,104,94),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,85,153,73,145),(66,86,154,74,146),(67,87,155,75,147),(68,88,156,76,148),(69,81,157,77,149),(70,82,158,78,150),(71,83,159,79,151),(72,84,160,80,152)], [(1,67,127,41),(2,42,128,68),(3,69,121,43),(4,44,122,70),(5,71,123,45),(6,46,124,72),(7,65,125,47),(8,48,126,66),(9,111,77,129),(10,130,78,112),(11,105,79,131),(12,132,80,106),(13,107,73,133),(14,134,74,108),(15,109,75,135),(16,136,76,110),(17,54,146,28),(18,29,147,55),(19,56,148,30),(20,31,149,49),(21,50,150,32),(22,25,151,51),(23,52,152,26),(24,27,145,53),(33,159,59,89),(34,90,60,160),(35,153,61,91),(36,92,62,154),(37,155,63,93),(38,94,64,156),(39,157,57,95),(40,96,58,158),(81,137,97,119),(82,120,98,138),(83,139,99,113),(84,114,100,140),(85,141,101,115),(86,116,102,142),(87,143,103,117),(88,118,104,144)], [(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)]])

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 20A ··· 20P 20Q ··· 20AF 40A ··· 40AF order 1 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C5 C8 C10 C10 C20 C40 D4 Q8 M4(2) C5×D4 C5×Q8 C5×M4(2) kernel C5×C4⋊C8 C4×C20 C2×C40 C2×C20 C4⋊C8 C20 C42 C2×C8 C2×C4 C4 C20 C20 C10 C4 C4 C2 # reps 1 1 2 4 4 8 4 8 16 32 1 1 2 4 4 8

Matrix representation of C5×C4⋊C8 in GL3(𝔽41) generated by

 10 0 0 0 18 0 0 0 18
,
 1 0 0 0 0 1 0 40 0
,
 38 0 0 0 28 23 0 23 13
G:=sub<GL(3,GF(41))| [10,0,0,0,18,0,0,0,18],[1,0,0,0,0,40,0,1,0],[38,0,0,0,28,23,0,23,13] >;

C5×C4⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes C_8
% in TeX

G:=Group("C5xC4:C8");
// GroupNames label

G:=SmallGroup(160,55);
// by ID

G=gap.SmallGroup(160,55);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,127,88]);
// Polycyclic

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

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