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## G = C32×D17order 306 = 2·32·17

### Direct product of C32 and D17

Aliases: C32×D17, C512C6, C17⋊(C3×C6), (C3×C51)⋊3C2, SmallGroup(306,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C32×D17
 Chief series C1 — C17 — C51 — C3×C51 — C32×D17
 Lower central C17 — C32×D17
 Upper central C1 — C32

Generators and relations for C32×D17
G = < a,b,c,d | a3=b3=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C32×D17
On 153 points
Generators in S153
(1 145 70)(2 146 71)(3 147 72)(4 148 73)(5 149 74)(6 150 75)(7 151 76)(8 152 77)(9 153 78)(10 137 79)(11 138 80)(12 139 81)(13 140 82)(14 141 83)(15 142 84)(16 143 85)(17 144 69)(18 105 87)(19 106 88)(20 107 89)(21 108 90)(22 109 91)(23 110 92)(24 111 93)(25 112 94)(26 113 95)(27 114 96)(28 115 97)(29 116 98)(30 117 99)(31 118 100)(32 119 101)(33 103 102)(34 104 86)(35 135 55)(36 136 56)(37 120 57)(38 121 58)(39 122 59)(40 123 60)(41 124 61)(42 125 62)(43 126 63)(44 127 64)(45 128 65)(46 129 66)(47 130 67)(48 131 68)(49 132 52)(50 133 53)(51 134 54)
(1 48 25)(2 49 26)(3 50 27)(4 51 28)(5 35 29)(6 36 30)(7 37 31)(8 38 32)(9 39 33)(10 40 34)(11 41 18)(12 42 19)(13 43 20)(14 44 21)(15 45 22)(16 46 23)(17 47 24)(52 95 71)(53 96 72)(54 97 73)(55 98 74)(56 99 75)(57 100 76)(58 101 77)(59 102 78)(60 86 79)(61 87 80)(62 88 81)(63 89 82)(64 90 83)(65 91 84)(66 92 85)(67 93 69)(68 94 70)(103 153 122)(104 137 123)(105 138 124)(106 139 125)(107 140 126)(108 141 127)(109 142 128)(110 143 129)(111 144 130)(112 145 131)(113 146 132)(114 147 133)(115 148 134)(116 149 135)(117 150 136)(118 151 120)(119 152 121)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(32 34)(35 43)(36 42)(37 41)(38 40)(44 51)(45 50)(46 49)(47 48)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(67 68)(69 70)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(120 124)(121 123)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)

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

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

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

90 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 17A ··· 17H 51A ··· 51BL order 1 2 3 ··· 3 6 ··· 6 17 ··· 17 51 ··· 51 size 1 17 1 ··· 1 17 ··· 17 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D17 C3×D17 kernel C32×D17 C3×C51 C3×D17 C51 C32 C3 # reps 1 1 8 8 8 64

Matrix representation of C32×D17 in GL3(𝔽103) generated by

 56 0 0 0 1 0 0 0 1
,
 56 0 0 0 46 0 0 0 46
,
 1 0 0 0 88 15 0 102 49
,
 102 0 0 0 48 100 0 81 55
G:=sub<GL(3,GF(103))| [56,0,0,0,1,0,0,0,1],[56,0,0,0,46,0,0,0,46],[1,0,0,0,88,102,0,15,49],[102,0,0,0,48,81,0,100,55] >;

C32×D17 in GAP, Magma, Sage, TeX

C_3^2\times D_{17}
% in TeX

G:=Group("C3^2xD17");
// GroupNames label

G:=SmallGroup(306,5);
// by ID

G=gap.SmallGroup(306,5);
# by ID

G:=PCGroup([4,-2,-3,-3,-17,4611]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^17=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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