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## G = C6×C9○He3order 486 = 2·35

### Direct product of C6 and C9○He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×C9○He3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C9○He3 — C6×C9○He3
 Lower central C1 — C3 — C6×C9○He3
 Upper central C1 — C3×C18 — C6×C9○He3

Generators and relations for C6×C9○He3
G = < a,b,c,d,e | a6=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 576 in 480 conjugacy classes, 432 normal (12 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, C33, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C32×C6, C32×C9, C3×He3, C3×3- 1+2, C9○He3, C32×C18, C6×He3, C6×3- 1+2, C2×C9○He3, C3×C9○He3, C6×C9○He3
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C9○He3, C34, C2×C9○He3, C33×C6, C3×C9○He3, C6×C9○He3

Smallest permutation representation of C6×C9○He3
On 162 points
Generators in S162
(1 122 29 112 41 102)(2 123 30 113 42 103)(3 124 31 114 43 104)(4 125 32 115 44 105)(5 126 33 116 45 106)(6 118 34 117 37 107)(7 119 35 109 38 108)(8 120 36 110 39 100)(9 121 28 111 40 101)(10 98 159 88 20 78)(11 99 160 89 21 79)(12 91 161 90 22 80)(13 92 162 82 23 81)(14 93 154 83 24 73)(15 94 155 84 25 74)(16 95 156 85 26 75)(17 96 157 86 27 76)(18 97 158 87 19 77)(46 137 66 127 56 147)(47 138 67 128 57 148)(48 139 68 129 58 149)(49 140 69 130 59 150)(50 141 70 131 60 151)(51 142 71 132 61 152)(52 143 72 133 62 153)(53 144 64 134 63 145)(54 136 65 135 55 146)
(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 65 75)(2 66 76)(3 67 77)(4 68 78)(5 69 79)(6 70 80)(7 71 81)(8 72 73)(9 64 74)(10 125 129)(11 126 130)(12 118 131)(13 119 132)(14 120 133)(15 121 134)(16 122 135)(17 123 127)(18 124 128)(19 104 138)(20 105 139)(21 106 140)(22 107 141)(23 108 142)(24 100 143)(25 101 144)(26 102 136)(27 103 137)(28 63 94)(29 55 95)(30 56 96)(31 57 97)(32 58 98)(33 59 99)(34 60 91)(35 61 92)(36 62 93)(37 50 90)(38 51 82)(39 52 83)(40 53 84)(41 54 85)(42 46 86)(43 47 87)(44 48 88)(45 49 89)(109 152 162)(110 153 154)(111 145 155)(112 146 156)(113 147 157)(114 148 158)(115 149 159)(116 150 160)(117 151 161)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)(145 151 148)(146 152 149)(147 153 150)(154 160 157)(155 161 158)(156 162 159)
(1 32 38)(2 33 39)(3 34 40)(4 35 41)(5 36 42)(6 28 43)(7 29 44)(8 30 45)(9 31 37)(10 156 23)(11 157 24)(12 158 25)(13 159 26)(14 160 27)(15 161 19)(16 162 20)(17 154 21)(18 155 22)(46 66 56)(47 67 57)(48 68 58)(49 69 59)(50 70 60)(51 71 61)(52 72 62)(53 64 63)(54 65 55)(73 99 86)(74 91 87)(75 92 88)(76 93 89)(77 94 90)(78 95 82)(79 96 83)(80 97 84)(81 98 85)(100 123 116)(101 124 117)(102 125 109)(103 126 110)(104 118 111)(105 119 112)(106 120 113)(107 121 114)(108 122 115)(127 147 137)(128 148 138)(129 149 139)(130 150 140)(131 151 141)(132 152 142)(133 153 143)(134 145 144)(135 146 136)

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

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

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

198 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 6A ··· 6H 6I ··· 6AF 9A ··· 9R 9S ··· 9BN 18A ··· 18R 18S ··· 18BN 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 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 C9○He3 C2×C9○He3 kernel C6×C9○He3 C3×C9○He3 C32×C18 C6×He3 C6×3- 1+2 C2×C9○He3 C32×C9 C3×He3 C3×3- 1+2 C9○He3 C6 C3 # reps 1 1 8 2 16 54 8 2 16 54 18 18

Matrix representation of C6×C9○He3 in GL4(𝔽19) generated by

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

C6×C9○He3 in GAP, Magma, Sage, TeX

C_6\times C_9\circ {\rm He}_3
% in TeX

G:=Group("C6xC9oHe3");
// GroupNames label

G:=SmallGroup(486,253);
// by ID

G=gap.SmallGroup(486,253);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,237]);
// Polycyclic

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

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