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## G = C17×3- 1+2order 459 = 33·17

### Direct product of C17 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C17×3- 1+2, C9⋊C51, C153⋊C3, C32.C51, C51.2C32, (C3×C51).C3, C3.2(C3×C51), SmallGroup(459,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C17×3- 1+2
 Chief series C1 — C3 — C51 — C153 — C17×3- 1+2
 Lower central C1 — C3 — C17×3- 1+2
 Upper central C1 — C51 — C17×3- 1+2

Generators and relations for C17×3- 1+2
G = < a,b,c | a17=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C17×3- 1+2
On 153 points
Generators in S153
(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)
(1 101 145 39 62 30 78 128 110)(2 102 146 40 63 31 79 129 111)(3 86 147 41 64 32 80 130 112)(4 87 148 42 65 33 81 131 113)(5 88 149 43 66 34 82 132 114)(6 89 150 44 67 18 83 133 115)(7 90 151 45 68 19 84 134 116)(8 91 152 46 52 20 85 135 117)(9 92 153 47 53 21 69 136 118)(10 93 137 48 54 22 70 120 119)(11 94 138 49 55 23 71 121 103)(12 95 139 50 56 24 72 122 104)(13 96 140 51 57 25 73 123 105)(14 97 141 35 58 26 74 124 106)(15 98 142 36 59 27 75 125 107)(16 99 143 37 60 28 76 126 108)(17 100 144 38 61 29 77 127 109)
(18 115 150)(19 116 151)(20 117 152)(21 118 153)(22 119 137)(23 103 138)(24 104 139)(25 105 140)(26 106 141)(27 107 142)(28 108 143)(29 109 144)(30 110 145)(31 111 146)(32 112 147)(33 113 148)(34 114 149)(52 91 135)(53 92 136)(54 93 120)(55 94 121)(56 95 122)(57 96 123)(58 97 124)(59 98 125)(60 99 126)(61 100 127)(62 101 128)(63 102 129)(64 86 130)(65 87 131)(66 88 132)(67 89 133)(68 90 134)

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

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

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

187 conjugacy classes

 class 1 3A 3B 3C 3D 9A ··· 9F 17A ··· 17P 51A ··· 51AF 51AG ··· 51BL 153A ··· 153CR order 1 3 3 3 3 9 ··· 9 17 ··· 17 51 ··· 51 51 ··· 51 153 ··· 153 size 1 1 1 3 3 3 ··· 3 1 ··· 1 1 ··· 1 3 ··· 3 3 ··· 3

187 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C17 C51 C51 3- 1+2 C17×3- 1+2 kernel C17×3- 1+2 C153 C3×C51 3- 1+2 C9 C32 C17 C1 # reps 1 6 2 16 96 32 2 32

Matrix representation of C17×3- 1+2 in GL3(𝔽307) generated by

 304 0 0 0 304 0 0 0 304
,
 306 288 0 204 1 289 187 177 0
,
 1 0 0 306 289 0 246 0 17
G:=sub<GL(3,GF(307))| [304,0,0,0,304,0,0,0,304],[306,204,187,288,1,177,0,289,0],[1,306,246,0,289,0,0,0,17] >;

C17×3- 1+2 in GAP, Magma, Sage, TeX

C_{17}\times 3_-^{1+2}
% in TeX

G:=Group("C17xES-(3,1)");
// GroupNames label

G:=SmallGroup(459,4);
// by ID

G=gap.SmallGroup(459,4);
# by ID

G:=PCGroup([4,-3,-3,-17,-3,612,1249]);
// Polycyclic

G:=Group<a,b,c|a^17=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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