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## G = C4×C5⋊C8order 160 = 25·5

### Direct product of C4 and C5⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C5⋊C8
G = < a,b,c | a4=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

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

C4×C5⋊C8 is a maximal subgroup of
D20⋊C8  Dic101C8  C40⋊C8  C20.31M4(2)  Dic5.23D8  Dic5.12Q16  C42.6F5  C42.12F5  Dic5.C42  C5⋊C88D4  Dic5⋊M4(2)  C23.(C2×F5)  D10.C42  D202C8  Dic10⋊C8  C20⋊M4(2)  C4⋊C4.7F5  Dic5.M4(2)  C20.M4(2)  Dic5.12M4(2)  C20.34M4(2)  C202M4(2)  C20.6M4(2)
C4×C5⋊C8 is a maximal quotient of
C42.4F5  C40⋊C8  Dic5⋊C16  C40.C8  C10.(C4⋊C8)

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C4 C8 C8 F5 C5⋊C8 C2×F5 D5⋊C8 C4×F5 kernel C4×C5⋊C8 C4×Dic5 C2×C5⋊C8 C5⋊C8 C2×Dic5 C2×C20 Dic5 C20 C2×C4 C4 C22 C2 C2 # reps 1 1 2 8 2 2 8 8 1 2 1 2 2

Matrix representation of C4×C5⋊C8 in GL5(𝔽41)

 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 0 0 40 0 1 0 0 40 0 0 1 0 40 0 0 0 1 40
,
 14 0 0 0 0 0 23 26 16 38 0 39 23 36 20 0 18 5 21 36 0 3 21 18 15

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[14,0,0,0,0,0,23,39,18,3,0,26,23,5,21,0,16,36,21,18,0,38,20,36,15] >;

C4×C5⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes C_8
% in TeX

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

G:=SmallGroup(160,75);
// by ID

G=gap.SmallGroup(160,75);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,86,2309,1169]);
// 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^3>;
// generators/relations

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