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## G = C6×C3.He3order 486 = 2·35

### Direct product of C6 and C3.He3

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

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 252 in 132 conjugacy classes, 72 normal (14 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3.He3, C32×C9, C3×3- 1+2, C2×C3.He3, C32×C18, C6×3- 1+2, C3×C3.He3, C6×C3.He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3.He3, C3×He3, C2×C3.He3, C6×He3, C3×C3.He3, C6×C3.He3

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

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

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

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

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

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

 8 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 4 0 0 0 0 4 0 0 9 0 6
,
 1 0 0 0 0 1 0 0 0 11 11 0 0 8 0 7
,
 11 0 0 0 0 7 15 0 0 0 12 11 0 11 18 0
G:=sub<GL(4,GF(19))| [8,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,4,0,9,0,0,4,0,0,0,0,6],[1,0,0,0,0,1,11,8,0,0,11,0,0,0,0,7],[11,0,0,0,0,7,0,11,0,15,12,18,0,0,11,0] >;

C6×C3.He3 in GAP, Magma, Sage, TeX

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

G:=Group("C6xC3.He3");
// GroupNames label

G:=SmallGroup(486,213);
// by ID

G=gap.SmallGroup(486,213);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,986,3250]);
// Polycyclic

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

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