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## G = C2×3- 1+2⋊C9order 486 = 2·35

### Direct product of C2 and 3- 1+2⋊C9

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×3- 1+2⋊C9
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — 3- 1+2⋊C9 — C2×3- 1+2⋊C9
 Lower central C1 — C3 — C32 — C2×3- 1+2⋊C9
 Upper central C1 — C3×C6 — C32×C6 — C2×3- 1+2⋊C9

Generators and relations for C2×3- 1+2⋊C9
G = < a,b,c,d | a2=b9=c3=d9=1, ab=ba, ac=ca, ad=da, cbc-1=b4, dbd-1=bc-1, dcd-1=b6c >

Subgroups: 198 in 82 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, 3- 1+2, C33, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C32⋊C9, C32×C9, C3×3- 1+2, C2×C32⋊C9, C32×C18, C6×3- 1+2, 3- 1+2⋊C9, C2×3- 1+2⋊C9
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C3≀C3, He3.C3, C3.He3, C2×C32⋊C9, C2×C3≀C3, C2×He3.C3, C2×C3.He3, 3- 1+2⋊C9, C2×3- 1+2⋊C9

Smallest permutation representation of C2×3- 1+2⋊C9
On 162 points
Generators in S162
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 109)(9 110)(10 87)(11 88)(12 89)(13 90)(14 82)(15 83)(16 84)(17 85)(18 86)(19 98)(20 99)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 105)(29 106)(30 107)(31 108)(32 100)(33 101)(34 102)(35 103)(36 104)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)(29 35 32)(30 33 36)(37 40 43)(39 45 42)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(64 70 67)(65 68 71)(73 76 79)(75 81 78)(82 85 88)(84 90 87)(91 97 94)(92 95 98)(100 106 103)(101 104 107)(109 115 112)(110 113 116)(118 121 124)(120 126 123)(128 134 131)(129 132 135)(136 139 142)(138 144 141)(145 151 148)(146 149 152)(154 157 160)(156 162 159)
(1 78 51 28 91 71 44 84 55)(2 76 52 29 98 72 45 82 56)(3 80 47 30 93 67 37 86 60)(4 81 54 31 94 65 38 87 58)(5 79 46 32 92 66 39 85 59)(6 74 50 33 96 70 40 89 63)(7 75 48 34 97 68 41 90 61)(8 73 49 35 95 69 42 88 62)(9 77 53 36 99 64 43 83 57)(10 139 114 162 135 108 24 146 119)(11 143 109 154 130 103 25 150 123)(12 144 116 155 131 101 26 151 121)(13 142 117 156 129 102 27 149 122)(14 137 112 157 133 106 19 153 126)(15 138 110 158 134 104 20 145 124)(16 136 111 159 132 105 21 152 125)(17 140 115 160 127 100 22 147 120)(18 141 113 161 128 107 23 148 118)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 6I ··· 6N 9A ··· 9R 9S ··· 9AJ 18A ··· 18R 18S ··· 18AJ order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 He3 3- 1+2 C2×He3 C2×3- 1+2 C3≀C3 He3.C3 C3.He3 C2×C3≀C3 C2×He3.C3 C2×C3.He3 kernel C2×3- 1+2⋊C9 3- 1+2⋊C9 C2×C32⋊C9 C32×C18 C6×3- 1+2 C32⋊C9 C32×C9 C3×3- 1+2 C2×3- 1+2 3- 1+2 C3×C6 C3×C6 C32 C32 C6 C6 C6 C3 C3 C3 # reps 1 1 4 2 2 4 2 2 18 18 2 4 2 4 6 6 6 6 6 6

Matrix representation of C2×3- 1+2⋊C9 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 7 0 0 0 0 7 9 0 0 1 12 1 0 0 18 0
,
 1 0 0 0 0 1 0 0 0 11 7 0 0 18 0 11
,
 16 0 0 0 0 9 17 0 0 0 10 6 0 0 15 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,7,1,0,0,9,12,18,0,0,1,0],[1,0,0,0,0,1,11,18,0,0,7,0,0,0,0,11],[16,0,0,0,0,9,0,0,0,17,10,15,0,0,6,0] >;

C2×3- 1+2⋊C9 in GAP, Magma, Sage, TeX

C_2\times 3_-^{1+2}\rtimes C_9
% in TeX

G:=Group("C2xES-(3,1):C9");
// GroupNames label

G:=SmallGroup(486,78);
// by ID

G=gap.SmallGroup(486,78);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,500,2169,735]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^4,d*b*d^-1=b*c^-1,d*c*d^-1=b^6*c>;
// generators/relations

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