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## G = C5×C8.4Q8order 320 = 26·5

### Direct product of C5 and C8.4Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×C8.4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C8.C4 — C5×C8.4Q8
 Lower central C1 — C2 — C4 — C8 — C5×C8.4Q8
 Upper central C1 — C20 — C2×C20 — C2×C40 — C5×C8.4Q8

Generators and relations for C5×C8.4Q8
G = < a,b,c,d | a5=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >

Smallest permutation representation of C5×C8.4Q8
On 160 points
Generators in S160
(1 71 100 35 123)(2 72 101 36 124)(3 73 102 37 125)(4 74 103 38 126)(5 75 104 39 127)(6 76 105 40 128)(7 77 106 41 113)(8 78 107 42 114)(9 79 108 43 115)(10 80 109 44 116)(11 65 110 45 117)(12 66 111 46 118)(13 67 112 47 119)(14 68 97 48 120)(15 69 98 33 121)(16 70 99 34 122)(17 138 55 83 151)(18 139 56 84 152)(19 140 57 85 153)(20 141 58 86 154)(21 142 59 87 155)(22 143 60 88 156)(23 144 61 89 157)(24 129 62 90 158)(25 130 63 91 159)(26 131 64 92 160)(27 132 49 93 145)(28 133 50 94 146)(29 134 51 95 147)(30 135 52 96 148)(31 136 53 81 149)(32 137 54 82 150)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)(65 75 69 79 73 67 77 71)(66 76 70 80 74 68 78 72)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 96)(97 107 101 111 105 99 109 103)(98 108 102 112 106 100 110 104)(113 123 117 127 121 115 125 119)(114 124 118 128 122 116 126 120)(129 131 133 135 137 139 141 143)(130 132 134 136 138 140 142 144)(145 147 149 151 153 155 157 159)(146 148 150 152 154 156 158 160)
(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)
(1 21 13 25 9 29 5 17)(2 20 14 24 10 28 6 32)(3 19 15 23 11 27 7 31)(4 18 16 22 12 26 8 30)(33 89 45 93 41 81 37 85)(34 88 46 92 42 96 38 84)(35 87 47 91 43 95 39 83)(36 86 48 90 44 94 40 82)(49 106 53 102 57 98 61 110)(50 105 54 101 58 97 62 109)(51 104 55 100 59 112 63 108)(52 103 56 99 60 111 64 107)(65 132 77 136 73 140 69 144)(66 131 78 135 74 139 70 143)(67 130 79 134 75 138 71 142)(68 129 80 133 76 137 72 141)(113 149 125 153 121 157 117 145)(114 148 126 152 122 156 118 160)(115 147 127 151 123 155 119 159)(116 146 128 150 124 154 120 158)

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

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

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

110 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 10H 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P 40Q ··· 40AF 80A ··· 80AF order 1 2 2 4 4 4 5 5 5 5 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 1 1 2 1 1 1 1 2 2 2 2 8 8 8 8 1 1 1 1 2 2 2 2 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2 8 ··· 8 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C4 C5 C10 C10 C20 Q8 D4 D8 Q16 C5×Q8 C5×D4 C8.4Q8 C5×D8 C5×Q16 C5×C8.4Q8 kernel C5×C8.4Q8 C5×C8.C4 C2×C80 C80 C8.4Q8 C8.C4 C2×C16 C16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C5 C4 C22 C1 # reps 1 2 1 4 4 8 4 16 1 1 2 2 4 4 8 8 8 32

Matrix representation of C5×C8.4Q8 in GL2(𝔽241) generated by

 98 0 0 98
,
 30 0 0 8
,
 126 0 0 44
,
 0 1 64 0
G:=sub<GL(2,GF(241))| [98,0,0,98],[30,0,0,8],[126,0,0,44],[0,64,1,0] >;

C5×C8.4Q8 in GAP, Magma, Sage, TeX

C_5\times C_8._4Q_8
% in TeX

G:=Group("C5xC8.4Q8");
// GroupNames label

G:=SmallGroup(320,173);
// by ID

G=gap.SmallGroup(320,173);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,708,2803,360,172,10085,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations

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