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## G = C5×C8.17D4order 320 = 26·5

### Direct product of C5 and C8.17D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×C8.17D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C8.C4 — C5×C8.17D4
 Lower central C1 — C2 — C4 — C8 — C5×C8.17D4
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×C8.17D4

Generators and relations for C5×C8.17D4
G = < a,b,c,d | a5=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C5×C8.17D4
On 160 points
Generators in S160
(1 56 149 34 123)(2 57 150 35 124)(3 58 151 36 125)(4 59 152 37 126)(5 60 153 38 127)(6 61 154 39 128)(7 62 155 40 113)(8 63 156 41 114)(9 64 157 42 115)(10 49 158 43 116)(11 50 159 44 117)(12 51 160 45 118)(13 52 145 46 119)(14 53 146 47 120)(15 54 147 48 121)(16 55 148 33 122)(17 99 132 90 73)(18 100 133 91 74)(19 101 134 92 75)(20 102 135 93 76)(21 103 136 94 77)(22 104 137 95 78)(23 105 138 96 79)(24 106 139 81 80)(25 107 140 82 65)(26 108 141 83 66)(27 109 142 84 67)(28 110 143 85 68)(29 111 144 86 69)(30 112 129 87 70)(31 97 130 88 71)(32 98 131 89 72)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)(65 79 77 75 73 71 69 67)(66 80 78 76 74 72 70 68)(81 95 93 91 89 87 85 83)(82 96 94 92 90 88 86 84)(97 111 109 107 105 103 101 99)(98 112 110 108 106 104 102 100)(113 127 125 123 121 119 117 115)(114 128 126 124 122 120 118 116)(129 143 141 139 137 135 133 131)(130 144 142 140 138 136 134 132)(145 159 157 155 153 151 149 147)(146 160 158 156 154 152 150 148)
(1 87 13 91 9 95 5 83)(2 90 6 86 10 82 14 94)(3 85 15 89 11 93 7 81)(4 88 8 84 12 96 16 92)(17 154 29 158 25 146 21 150)(18 157 22 153 26 149 30 145)(19 152 31 156 27 160 23 148)(20 155 24 151 28 147 32 159)(33 101 37 97 41 109 45 105)(34 112 46 100 42 104 38 108)(35 99 39 111 43 107 47 103)(36 110 48 98 44 102 40 106)(49 65 53 77 57 73 61 69)(50 76 62 80 58 68 54 72)(51 79 55 75 59 71 63 67)(52 74 64 78 60 66 56 70)(113 139 125 143 121 131 117 135)(114 142 118 138 122 134 126 130)(115 137 127 141 123 129 119 133)(116 140 120 136 124 132 128 144)
(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)| (1,56,149,34,123)(2,57,150,35,124)(3,58,151,36,125)(4,59,152,37,126)(5,60,153,38,127)(6,61,154,39,128)(7,62,155,40,113)(8,63,156,41,114)(9,64,157,42,115)(10,49,158,43,116)(11,50,159,44,117)(12,51,160,45,118)(13,52,145,46,119)(14,53,146,47,120)(15,54,147,48,121)(16,55,148,33,122)(17,99,132,90,73)(18,100,133,91,74)(19,101,134,92,75)(20,102,135,93,76)(21,103,136,94,77)(22,104,137,95,78)(23,105,138,96,79)(24,106,139,81,80)(25,107,140,82,65)(26,108,141,83,66)(27,109,142,84,67)(28,110,143,85,68)(29,111,144,86,69)(30,112,129,87,70)(31,97,130,88,71)(32,98,131,89,72), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84)(97,111,109,107,105,103,101,99)(98,112,110,108,106,104,102,100)(113,127,125,123,121,119,117,115)(114,128,126,124,122,120,118,116)(129,143,141,139,137,135,133,131)(130,144,142,140,138,136,134,132)(145,159,157,155,153,151,149,147)(146,160,158,156,154,152,150,148), (1,87,13,91,9,95,5,83)(2,90,6,86,10,82,14,94)(3,85,15,89,11,93,7,81)(4,88,8,84,12,96,16,92)(17,154,29,158,25,146,21,150)(18,157,22,153,26,149,30,145)(19,152,31,156,27,160,23,148)(20,155,24,151,28,147,32,159)(33,101,37,97,41,109,45,105)(34,112,46,100,42,104,38,108)(35,99,39,111,43,107,47,103)(36,110,48,98,44,102,40,106)(49,65,53,77,57,73,61,69)(50,76,62,80,58,68,54,72)(51,79,55,75,59,71,63,67)(52,74,64,78,60,66,56,70)(113,139,125,143,121,131,117,135)(114,142,118,138,122,134,126,130)(115,137,127,141,123,129,119,133)(116,140,120,136,124,132,128,144), (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( (1,56,149,34,123)(2,57,150,35,124)(3,58,151,36,125)(4,59,152,37,126)(5,60,153,38,127)(6,61,154,39,128)(7,62,155,40,113)(8,63,156,41,114)(9,64,157,42,115)(10,49,158,43,116)(11,50,159,44,117)(12,51,160,45,118)(13,52,145,46,119)(14,53,146,47,120)(15,54,147,48,121)(16,55,148,33,122)(17,99,132,90,73)(18,100,133,91,74)(19,101,134,92,75)(20,102,135,93,76)(21,103,136,94,77)(22,104,137,95,78)(23,105,138,96,79)(24,106,139,81,80)(25,107,140,82,65)(26,108,141,83,66)(27,109,142,84,67)(28,110,143,85,68)(29,111,144,86,69)(30,112,129,87,70)(31,97,130,88,71)(32,98,131,89,72), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84)(97,111,109,107,105,103,101,99)(98,112,110,108,106,104,102,100)(113,127,125,123,121,119,117,115)(114,128,126,124,122,120,118,116)(129,143,141,139,137,135,133,131)(130,144,142,140,138,136,134,132)(145,159,157,155,153,151,149,147)(146,160,158,156,154,152,150,148), (1,87,13,91,9,95,5,83)(2,90,6,86,10,82,14,94)(3,85,15,89,11,93,7,81)(4,88,8,84,12,96,16,92)(17,154,29,158,25,146,21,150)(18,157,22,153,26,149,30,145)(19,152,31,156,27,160,23,148)(20,155,24,151,28,147,32,159)(33,101,37,97,41,109,45,105)(34,112,46,100,42,104,38,108)(35,99,39,111,43,107,47,103)(36,110,48,98,44,102,40,106)(49,65,53,77,57,73,61,69)(50,76,62,80,58,68,54,72)(51,79,55,75,59,71,63,67)(52,74,64,78,60,66,56,70)(113,139,125,143,121,131,117,135)(114,142,118,138,122,134,126,130)(115,137,127,141,123,129,119,133)(116,140,120,136,124,132,128,144), (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([[(1,56,149,34,123),(2,57,150,35,124),(3,58,151,36,125),(4,59,152,37,126),(5,60,153,38,127),(6,61,154,39,128),(7,62,155,40,113),(8,63,156,41,114),(9,64,157,42,115),(10,49,158,43,116),(11,50,159,44,117),(12,51,160,45,118),(13,52,145,46,119),(14,53,146,47,120),(15,54,147,48,121),(16,55,148,33,122),(17,99,132,90,73),(18,100,133,91,74),(19,101,134,92,75),(20,102,135,93,76),(21,103,136,94,77),(22,104,137,95,78),(23,105,138,96,79),(24,106,139,81,80),(25,107,140,82,65),(26,108,141,83,66),(27,109,142,84,67),(28,110,143,85,68),(29,111,144,86,69),(30,112,129,87,70),(31,97,130,88,71),(32,98,131,89,72)], [(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20),(33,47,45,43,41,39,37,35),(34,48,46,44,42,40,38,36),(49,63,61,59,57,55,53,51),(50,64,62,60,58,56,54,52),(65,79,77,75,73,71,69,67),(66,80,78,76,74,72,70,68),(81,95,93,91,89,87,85,83),(82,96,94,92,90,88,86,84),(97,111,109,107,105,103,101,99),(98,112,110,108,106,104,102,100),(113,127,125,123,121,119,117,115),(114,128,126,124,122,120,118,116),(129,143,141,139,137,135,133,131),(130,144,142,140,138,136,134,132),(145,159,157,155,153,151,149,147),(146,160,158,156,154,152,150,148)], [(1,87,13,91,9,95,5,83),(2,90,6,86,10,82,14,94),(3,85,15,89,11,93,7,81),(4,88,8,84,12,96,16,92),(17,154,29,158,25,146,21,150),(18,157,22,153,26,149,30,145),(19,152,31,156,27,160,23,148),(20,155,24,151,28,147,32,159),(33,101,37,97,41,109,45,105),(34,112,46,100,42,104,38,108),(35,99,39,111,43,107,47,103),(36,110,48,98,44,102,40,106),(49,65,53,77,57,73,61,69),(50,76,62,80,58,68,54,72),(51,79,55,75,59,71,63,67),(52,74,64,78,60,66,56,70),(113,139,125,143,121,131,117,135),(114,142,118,138,122,134,126,130),(115,137,127,141,123,129,119,133),(116,140,120,136,124,132,128,144)], [(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)]])

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 D4 D8 SD16 C5×D4 C5×D4 C5×D8 C5×SD16 C8.17D4 C5×C8.17D4 kernel C5×C8.17D4 C5×C8.C4 C5×M5(2) C10×Q16 C5×Q16 C8.17D4 C8.C4 M5(2) C2×Q16 Q16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×C8.17D4 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 11 230 169 57 11 11 184 169 0 0 11 11 0 0 230 11
,
 89 22 23 112 165 117 137 4 0 2 124 165 239 0 22 152
,
 168 185 201 109 72 167 67 13 118 131 76 32 131 123 160 71
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[11,11,0,0,230,11,0,0,169,184,11,230,57,169,11,11],[89,165,0,239,22,117,2,0,23,137,124,22,112,4,165,152],[168,72,118,131,185,167,131,123,201,67,76,160,109,13,32,71] >;

C5×C8.17D4 in GAP, Magma, Sage, TeX

C_5\times C_8._{17}D_4
% in TeX

G:=Group("C5xC8.17D4");
// GroupNames label

G:=SmallGroup(320,167);
// by ID

G=gap.SmallGroup(320,167);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,3650,136,3511,172,10085,5052,124]);
// Polycyclic

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

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