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## G = C3×C6×3- 1+2order 486 = 2·35

### Direct product of C3×C6 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×C6×3- 1+2
 Chief series C1 — C3 — C32 — C33 — C34 — C32×3- 1+2 — C3×C6×3- 1+2
 Lower central C1 — C3 — C3×C6×3- 1+2
 Upper central C1 — C32×C6 — C3×C6×3- 1+2

Generators and relations for C3×C6×3- 1+2
G = < a,b,c,d | a3=b6=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 792 in 576 conjugacy classes, 468 normal (10 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C18, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C33, C33, C3×C18, C2×3- 1+2, C32×C6, C32×C6, C32×C6, C32×C9, C3×3- 1+2, C34, C32×C18, C6×3- 1+2, C33×C6, C32×3- 1+2, C3×C6×3- 1+2
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C34, C6×3- 1+2, C33×C6, C32×3- 1+2, C3×C6×3- 1+2

Smallest permutation representation of C3×C6×3- 1+2
On 162 points
Generators in S162
(1 41 118)(2 42 119)(3 43 120)(4 44 121)(5 45 122)(6 37 123)(7 38 124)(8 39 125)(9 40 126)(10 56 101)(11 57 102)(12 58 103)(13 59 104)(14 60 105)(15 61 106)(16 62 107)(17 63 108)(18 55 100)(19 80 28)(20 81 29)(21 73 30)(22 74 31)(23 75 32)(24 76 33)(25 77 34)(26 78 35)(27 79 36)(46 95 82)(47 96 83)(48 97 84)(49 98 85)(50 99 86)(51 91 87)(52 92 88)(53 93 89)(54 94 90)(64 115 129)(65 116 130)(66 117 131)(67 109 132)(68 110 133)(69 111 134)(70 112 135)(71 113 127)(72 114 128)(136 159 149)(137 160 150)(138 161 151)(139 162 152)(140 154 153)(141 155 145)(142 156 146)(143 157 147)(144 158 148)
(1 50 22 117 100 141)(2 51 23 109 101 142)(3 52 24 110 102 143)(4 53 25 111 103 144)(5 54 26 112 104 136)(6 46 27 113 105 137)(7 47 19 114 106 138)(8 48 20 115 107 139)(9 49 21 116 108 140)(10 156 42 91 75 132)(11 157 43 92 76 133)(12 158 44 93 77 134)(13 159 45 94 78 135)(14 160 37 95 79 127)(15 161 38 96 80 128)(16 162 39 97 81 129)(17 154 40 98 73 130)(18 155 41 99 74 131)(28 72 61 151 124 83)(29 64 62 152 125 84)(30 65 63 153 126 85)(31 66 55 145 118 86)(32 67 56 146 119 87)(33 68 57 147 120 88)(34 69 58 148 121 89)(35 70 59 149 122 90)(36 71 60 150 123 82)
(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)(154 155 156 157 158 159 160 161 162)
(1 77 61)(2 75 56)(3 73 60)(4 80 55)(5 78 59)(6 76 63)(7 74 58)(8 81 62)(9 79 57)(10 119 23)(11 126 27)(12 124 22)(13 122 26)(14 120 21)(15 118 25)(16 125 20)(17 123 24)(18 121 19)(28 100 44)(29 107 39)(30 105 43)(31 103 38)(32 101 42)(33 108 37)(34 106 41)(35 104 45)(36 102 40)(46 133 153)(47 131 148)(48 129 152)(49 127 147)(50 134 151)(51 132 146)(52 130 150)(53 128 145)(54 135 149)(64 139 97)(65 137 92)(66 144 96)(67 142 91)(68 140 95)(69 138 99)(70 136 94)(71 143 98)(72 141 93)(82 110 154)(83 117 158)(84 115 162)(85 113 157)(86 111 161)(87 109 156)(88 116 160)(89 114 155)(90 112 159)

G:=sub<Sym(162)| (1,41,118)(2,42,119)(3,43,120)(4,44,121)(5,45,122)(6,37,123)(7,38,124)(8,39,125)(9,40,126)(10,56,101)(11,57,102)(12,58,103)(13,59,104)(14,60,105)(15,61,106)(16,62,107)(17,63,108)(18,55,100)(19,80,28)(20,81,29)(21,73,30)(22,74,31)(23,75,32)(24,76,33)(25,77,34)(26,78,35)(27,79,36)(46,95,82)(47,96,83)(48,97,84)(49,98,85)(50,99,86)(51,91,87)(52,92,88)(53,93,89)(54,94,90)(64,115,129)(65,116,130)(66,117,131)(67,109,132)(68,110,133)(69,111,134)(70,112,135)(71,113,127)(72,114,128)(136,159,149)(137,160,150)(138,161,151)(139,162,152)(140,154,153)(141,155,145)(142,156,146)(143,157,147)(144,158,148), (1,50,22,117,100,141)(2,51,23,109,101,142)(3,52,24,110,102,143)(4,53,25,111,103,144)(5,54,26,112,104,136)(6,46,27,113,105,137)(7,47,19,114,106,138)(8,48,20,115,107,139)(9,49,21,116,108,140)(10,156,42,91,75,132)(11,157,43,92,76,133)(12,158,44,93,77,134)(13,159,45,94,78,135)(14,160,37,95,79,127)(15,161,38,96,80,128)(16,162,39,97,81,129)(17,154,40,98,73,130)(18,155,41,99,74,131)(28,72,61,151,124,83)(29,64,62,152,125,84)(30,65,63,153,126,85)(31,66,55,145,118,86)(32,67,56,146,119,87)(33,68,57,147,120,88)(34,69,58,148,121,89)(35,70,59,149,122,90)(36,71,60,150,123,82), (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)(154,155,156,157,158,159,160,161,162), (1,77,61)(2,75,56)(3,73,60)(4,80,55)(5,78,59)(6,76,63)(7,74,58)(8,81,62)(9,79,57)(10,119,23)(11,126,27)(12,124,22)(13,122,26)(14,120,21)(15,118,25)(16,125,20)(17,123,24)(18,121,19)(28,100,44)(29,107,39)(30,105,43)(31,103,38)(32,101,42)(33,108,37)(34,106,41)(35,104,45)(36,102,40)(46,133,153)(47,131,148)(48,129,152)(49,127,147)(50,134,151)(51,132,146)(52,130,150)(53,128,145)(54,135,149)(64,139,97)(65,137,92)(66,144,96)(67,142,91)(68,140,95)(69,138,99)(70,136,94)(71,143,98)(72,141,93)(82,110,154)(83,117,158)(84,115,162)(85,113,157)(86,111,161)(87,109,156)(88,116,160)(89,114,155)(90,112,159)>;

G:=Group( (1,41,118)(2,42,119)(3,43,120)(4,44,121)(5,45,122)(6,37,123)(7,38,124)(8,39,125)(9,40,126)(10,56,101)(11,57,102)(12,58,103)(13,59,104)(14,60,105)(15,61,106)(16,62,107)(17,63,108)(18,55,100)(19,80,28)(20,81,29)(21,73,30)(22,74,31)(23,75,32)(24,76,33)(25,77,34)(26,78,35)(27,79,36)(46,95,82)(47,96,83)(48,97,84)(49,98,85)(50,99,86)(51,91,87)(52,92,88)(53,93,89)(54,94,90)(64,115,129)(65,116,130)(66,117,131)(67,109,132)(68,110,133)(69,111,134)(70,112,135)(71,113,127)(72,114,128)(136,159,149)(137,160,150)(138,161,151)(139,162,152)(140,154,153)(141,155,145)(142,156,146)(143,157,147)(144,158,148), (1,50,22,117,100,141)(2,51,23,109,101,142)(3,52,24,110,102,143)(4,53,25,111,103,144)(5,54,26,112,104,136)(6,46,27,113,105,137)(7,47,19,114,106,138)(8,48,20,115,107,139)(9,49,21,116,108,140)(10,156,42,91,75,132)(11,157,43,92,76,133)(12,158,44,93,77,134)(13,159,45,94,78,135)(14,160,37,95,79,127)(15,161,38,96,80,128)(16,162,39,97,81,129)(17,154,40,98,73,130)(18,155,41,99,74,131)(28,72,61,151,124,83)(29,64,62,152,125,84)(30,65,63,153,126,85)(31,66,55,145,118,86)(32,67,56,146,119,87)(33,68,57,147,120,88)(34,69,58,148,121,89)(35,70,59,149,122,90)(36,71,60,150,123,82), (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)(154,155,156,157,158,159,160,161,162), (1,77,61)(2,75,56)(3,73,60)(4,80,55)(5,78,59)(6,76,63)(7,74,58)(8,81,62)(9,79,57)(10,119,23)(11,126,27)(12,124,22)(13,122,26)(14,120,21)(15,118,25)(16,125,20)(17,123,24)(18,121,19)(28,100,44)(29,107,39)(30,105,43)(31,103,38)(32,101,42)(33,108,37)(34,106,41)(35,104,45)(36,102,40)(46,133,153)(47,131,148)(48,129,152)(49,127,147)(50,134,151)(51,132,146)(52,130,150)(53,128,145)(54,135,149)(64,139,97)(65,137,92)(66,144,96)(67,142,91)(68,140,95)(69,138,99)(70,136,94)(71,143,98)(72,141,93)(82,110,154)(83,117,158)(84,115,162)(85,113,157)(86,111,161)(87,109,156)(88,116,160)(89,114,155)(90,112,159) );

G=PermutationGroup([[(1,41,118),(2,42,119),(3,43,120),(4,44,121),(5,45,122),(6,37,123),(7,38,124),(8,39,125),(9,40,126),(10,56,101),(11,57,102),(12,58,103),(13,59,104),(14,60,105),(15,61,106),(16,62,107),(17,63,108),(18,55,100),(19,80,28),(20,81,29),(21,73,30),(22,74,31),(23,75,32),(24,76,33),(25,77,34),(26,78,35),(27,79,36),(46,95,82),(47,96,83),(48,97,84),(49,98,85),(50,99,86),(51,91,87),(52,92,88),(53,93,89),(54,94,90),(64,115,129),(65,116,130),(66,117,131),(67,109,132),(68,110,133),(69,111,134),(70,112,135),(71,113,127),(72,114,128),(136,159,149),(137,160,150),(138,161,151),(139,162,152),(140,154,153),(141,155,145),(142,156,146),(143,157,147),(144,158,148)], [(1,50,22,117,100,141),(2,51,23,109,101,142),(3,52,24,110,102,143),(4,53,25,111,103,144),(5,54,26,112,104,136),(6,46,27,113,105,137),(7,47,19,114,106,138),(8,48,20,115,107,139),(9,49,21,116,108,140),(10,156,42,91,75,132),(11,157,43,92,76,133),(12,158,44,93,77,134),(13,159,45,94,78,135),(14,160,37,95,79,127),(15,161,38,96,80,128),(16,162,39,97,81,129),(17,154,40,98,73,130),(18,155,41,99,74,131),(28,72,61,151,124,83),(29,64,62,152,125,84),(30,65,63,153,126,85),(31,66,55,145,118,86),(32,67,56,146,119,87),(33,68,57,147,120,88),(34,69,58,148,121,89),(35,70,59,149,122,90),(36,71,60,150,123,82)], [(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),(154,155,156,157,158,159,160,161,162)], [(1,77,61),(2,75,56),(3,73,60),(4,80,55),(5,78,59),(6,76,63),(7,74,58),(8,81,62),(9,79,57),(10,119,23),(11,126,27),(12,124,22),(13,122,26),(14,120,21),(15,118,25),(16,125,20),(17,123,24),(18,121,19),(28,100,44),(29,107,39),(30,105,43),(31,103,38),(32,101,42),(33,108,37),(34,106,41),(35,104,45),(36,102,40),(46,133,153),(47,131,148),(48,129,152),(49,127,147),(50,134,151),(51,132,146),(52,130,150),(53,128,145),(54,135,149),(64,139,97),(65,137,92),(66,144,96),(67,142,91),(68,140,95),(69,138,99),(70,136,94),(71,143,98),(72,141,93),(82,110,154),(83,117,158),(84,115,162),(85,113,157),(86,111,161),(87,109,156),(88,116,160),(89,114,155),(90,112,159)]])

198 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3AR 6A ··· 6Z 6AA ··· 6AR 9A ··· 9BB 18A ··· 18BB order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 3- 1+2 C2×3- 1+2 kernel C3×C6×3- 1+2 C32×3- 1+2 C32×C18 C6×3- 1+2 C33×C6 C32×C9 C3×3- 1+2 C34 C3×C6 C32 # reps 1 1 6 72 2 6 72 2 18 18

Matrix representation of C3×C6×3- 1+2 in GL5(𝔽19)

 7 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 18 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 7 0 0 0 0 0 11 0 0 0 0 0 17 1 17 0 0 8 0 14 0 0 6 0 2
,
 11 0 0 0 0 0 11 0 0 0 0 0 7 0 5 0 0 0 1 7 0 0 0 0 11

G:=sub<GL(5,GF(19))| [7,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[18,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,0,0,0,0,0,11,0,0,0,0,0,17,8,6,0,0,1,0,0,0,0,17,14,2],[11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,5,7,11] >;

C3×C6×3- 1+2 in GAP, Magma, Sage, TeX

C_3\times C_6\times 3_-^{1+2}
% in TeX

G:=Group("C3xC6xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,252);
// by ID

G=gap.SmallGroup(486,252);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1520]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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