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## G = C6×He3.C3order 486 = 2·35

### Direct product of C6 and He3.C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×He3.C3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×He3.C3 — C6×He3.C3
 Lower central C1 — C3 — C32 — C6×He3.C3
 Upper central C1 — C3×C6 — C32×C6 — C6×He3.C3

Generators and relations for C6×He3.C3
G = < a,b,c,d,e | a6=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Subgroups: 360 in 144 conjugacy classes, 72 normal (16 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, He3, 3- 1+2, 3- 1+2, C33, C33, C3×C18, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C32×C6, He3.C3, C32×C9, C3×He3, C3×3- 1+2, C2×He3.C3, C32×C18, C6×He3, C6×3- 1+2, C3×He3.C3, C6×He3.C3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, He3.C3, C3×He3, C2×He3.C3, C6×He3, C3×He3.C3, C6×He3.C3

Smallest permutation representation of C6×He3.C3
On 162 points
Generators in S162
(1 122 28 112 41 102)(2 123 29 113 42 103)(3 124 30 114 43 104)(4 125 31 115 44 105)(5 126 32 116 45 106)(6 118 33 117 37 107)(7 119 34 109 38 108)(8 120 35 110 39 100)(9 121 36 111 40 101)(10 95 162 88 26 81)(11 96 154 89 27 73)(12 97 155 90 19 74)(13 98 156 82 20 75)(14 99 157 83 21 76)(15 91 158 84 22 77)(16 92 159 85 23 78)(17 93 160 86 24 79)(18 94 161 87 25 80)(46 144 65 127 63 146)(47 136 66 128 55 147)(48 137 67 129 56 148)(49 138 68 130 57 149)(50 139 69 131 58 150)(51 140 70 132 59 151)(52 141 71 133 60 152)(53 142 72 134 61 153)(54 143 64 135 62 145)
(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(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)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 160 157)(155 161 158)(156 162 159)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 157 160)(155 158 161)(156 159 162)
(1 78 64)(2 76 65)(3 74 66)(4 81 67)(5 79 68)(6 77 69)(7 75 70)(8 73 71)(9 80 72)(10 129 125)(11 133 120)(12 128 124)(13 132 119)(14 127 123)(15 131 118)(16 135 122)(17 130 126)(18 134 121)(19 136 104)(20 140 108)(21 144 103)(22 139 107)(23 143 102)(24 138 106)(25 142 101)(26 137 105)(27 141 100)(28 92 62)(29 99 63)(30 97 55)(31 95 56)(32 93 57)(33 91 58)(34 98 59)(35 96 60)(36 94 61)(37 84 50)(38 82 51)(39 89 52)(40 87 53)(41 85 54)(42 83 46)(43 90 47)(44 88 48)(45 86 49)(109 156 151)(110 154 152)(111 161 153)(112 159 145)(113 157 146)(114 155 147)(115 162 148)(116 160 149)(117 158 150)
(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)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 3O ··· 3T 6A ··· 6H 6I ··· 6N 6O ··· 6T 9A ··· 9R 9S ··· 9AD 18A ··· 18R 18S ··· 18AD order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9 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 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 He3 C2×He3 He3.C3 C2×He3.C3 kernel C6×He3.C3 C3×He3.C3 C2×He3.C3 C32×C18 C6×He3 C6×3- 1+2 He3.C3 C32×C9 C3×He3 C3×3- 1+2 C3×C6 C32 C6 C3 # reps 1 1 18 2 2 4 18 2 2 4 6 6 18 18

Matrix representation of C6×He3.C3 in GL4(𝔽19) generated by

 12 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 0 0 0 0 11 0 0 7 6 7
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 11 0 0 0 0 0 1 0 0 18 4 10 0 10 8 15
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 3 7 5
G:=sub<GL(4,GF(19))| [12,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,7,0,0,11,6,0,0,0,7],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[11,0,0,0,0,0,18,10,0,1,4,8,0,0,10,15],[1,0,0,0,0,16,0,3,0,0,16,7,0,0,0,5] >;

C6×He3.C3 in GAP, Magma, Sage, TeX

C_6\times {\rm He}_3.C_3
% in TeX

G:=Group("C6xHe3.C3");
// GroupNames label

G:=SmallGroup(486,211);
// by ID

G=gap.SmallGroup(486,211);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,500,3250]);
// Polycyclic

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

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