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## G = C2×C9.6He3order 486 = 2·35

### Direct product of C2 and C9.6He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C9.6He3
 Chief series C1 — C3 — C9 — C3×C9 — C9○He3 — C9.6He3 — C2×C9.6He3
 Lower central C1 — C3 — C32 — C2×C9.6He3
 Upper central C1 — C18 — C3×C18 — C2×C9.6He3

Generators and relations for C2×C9.6He3
G = < a,b,c,d,e | a2=b9=c3=d3=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b6c, ece-1=b6cd-1, ede-1=b3d >

Smallest permutation representation of C2×C9.6He3
On 162 points
Generators in S162
(1 51)(2 52)(3 53)(4 54)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 109)(80 110)(81 111)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 161)(97 162)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106)(83 86 89 92 95 98 101 104 107)(84 87 90 93 96 99 102 105 108)(109 112 115 118 121 124 127 130 133)(110 113 116 119 122 125 128 131 134)(111 114 117 120 123 126 129 132 135)(136 139 142 145 148 151 154 157 160)(137 140 143 146 149 152 155 158 161)(138 141 144 147 150 153 156 159 162)
(1 124 88)(2 134 98)(3 135 90)(4 127 91)(5 110 101)(6 111 93)(7 130 94)(8 113 104)(9 114 96)(10 133 97)(11 116 107)(12 117 99)(13 109 100)(14 119 83)(15 120 102)(16 112 103)(17 122 86)(18 123 105)(19 115 106)(20 125 89)(21 126 108)(22 118 82)(23 128 92)(24 129 84)(25 121 85)(26 131 95)(27 132 87)(28 80 139)(29 81 158)(30 73 159)(31 56 142)(32 57 161)(33 76 162)(34 59 145)(35 60 137)(36 79 138)(37 62 148)(38 63 140)(39 55 141)(40 65 151)(41 66 143)(42 58 144)(43 68 154)(44 69 146)(45 61 147)(46 71 157)(47 72 149)(48 64 150)(49 74 160)(50 75 152)(51 67 153)(52 77 136)(53 78 155)(54 70 156)
(2 11 20)(3 21 12)(5 14 23)(6 24 15)(8 17 26)(9 27 18)(28 37 46)(29 47 38)(31 40 49)(32 50 41)(34 43 52)(35 53 44)(55 64 73)(56 74 65)(58 67 76)(59 77 68)(61 70 79)(62 80 71)(82 100 91)(84 93 102)(85 103 94)(87 96 105)(88 106 97)(90 99 108)(109 118 127)(110 128 119)(112 121 130)(113 131 122)(115 124 133)(116 134 125)(137 146 155)(138 156 147)(140 149 158)(141 159 150)(143 152 161)(144 162 153)
(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)| (1,51)(2,52)(3,53)(4,54)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,109)(80,110)(81,111)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (1,124,88)(2,134,98)(3,135,90)(4,127,91)(5,110,101)(6,111,93)(7,130,94)(8,113,104)(9,114,96)(10,133,97)(11,116,107)(12,117,99)(13,109,100)(14,119,83)(15,120,102)(16,112,103)(17,122,86)(18,123,105)(19,115,106)(20,125,89)(21,126,108)(22,118,82)(23,128,92)(24,129,84)(25,121,85)(26,131,95)(27,132,87)(28,80,139)(29,81,158)(30,73,159)(31,56,142)(32,57,161)(33,76,162)(34,59,145)(35,60,137)(36,79,138)(37,62,148)(38,63,140)(39,55,141)(40,65,151)(41,66,143)(42,58,144)(43,68,154)(44,69,146)(45,61,147)(46,71,157)(47,72,149)(48,64,150)(49,74,160)(50,75,152)(51,67,153)(52,77,136)(53,78,155)(54,70,156), (2,11,20)(3,21,12)(5,14,23)(6,24,15)(8,17,26)(9,27,18)(28,37,46)(29,47,38)(31,40,49)(32,50,41)(34,43,52)(35,53,44)(55,64,73)(56,74,65)(58,67,76)(59,77,68)(61,70,79)(62,80,71)(82,100,91)(84,93,102)(85,103,94)(87,96,105)(88,106,97)(90,99,108)(109,118,127)(110,128,119)(112,121,130)(113,131,122)(115,124,133)(116,134,125)(137,146,155)(138,156,147)(140,149,158)(141,159,150)(143,152,161)(144,162,153), (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( (1,51)(2,52)(3,53)(4,54)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,109)(80,110)(81,111)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (1,124,88)(2,134,98)(3,135,90)(4,127,91)(5,110,101)(6,111,93)(7,130,94)(8,113,104)(9,114,96)(10,133,97)(11,116,107)(12,117,99)(13,109,100)(14,119,83)(15,120,102)(16,112,103)(17,122,86)(18,123,105)(19,115,106)(20,125,89)(21,126,108)(22,118,82)(23,128,92)(24,129,84)(25,121,85)(26,131,95)(27,132,87)(28,80,139)(29,81,158)(30,73,159)(31,56,142)(32,57,161)(33,76,162)(34,59,145)(35,60,137)(36,79,138)(37,62,148)(38,63,140)(39,55,141)(40,65,151)(41,66,143)(42,58,144)(43,68,154)(44,69,146)(45,61,147)(46,71,157)(47,72,149)(48,64,150)(49,74,160)(50,75,152)(51,67,153)(52,77,136)(53,78,155)(54,70,156), (2,11,20)(3,21,12)(5,14,23)(6,24,15)(8,17,26)(9,27,18)(28,37,46)(29,47,38)(31,40,49)(32,50,41)(34,43,52)(35,53,44)(55,64,73)(56,74,65)(58,67,76)(59,77,68)(61,70,79)(62,80,71)(82,100,91)(84,93,102)(85,103,94)(87,96,105)(88,106,97)(90,99,108)(109,118,127)(110,128,119)(112,121,130)(113,131,122)(115,124,133)(116,134,125)(137,146,155)(138,156,147)(140,149,158)(141,159,150)(143,152,161)(144,162,153), (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([[(1,51),(2,52),(3,53),(4,54),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(61,118),(62,119),(63,120),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,109),(80,110),(81,111),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,161),(97,162),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106),(83,86,89,92,95,98,101,104,107),(84,87,90,93,96,99,102,105,108),(109,112,115,118,121,124,127,130,133),(110,113,116,119,122,125,128,131,134),(111,114,117,120,123,126,129,132,135),(136,139,142,145,148,151,154,157,160),(137,140,143,146,149,152,155,158,161),(138,141,144,147,150,153,156,159,162)], [(1,124,88),(2,134,98),(3,135,90),(4,127,91),(5,110,101),(6,111,93),(7,130,94),(8,113,104),(9,114,96),(10,133,97),(11,116,107),(12,117,99),(13,109,100),(14,119,83),(15,120,102),(16,112,103),(17,122,86),(18,123,105),(19,115,106),(20,125,89),(21,126,108),(22,118,82),(23,128,92),(24,129,84),(25,121,85),(26,131,95),(27,132,87),(28,80,139),(29,81,158),(30,73,159),(31,56,142),(32,57,161),(33,76,162),(34,59,145),(35,60,137),(36,79,138),(37,62,148),(38,63,140),(39,55,141),(40,65,151),(41,66,143),(42,58,144),(43,68,154),(44,69,146),(45,61,147),(46,71,157),(47,72,149),(48,64,150),(49,74,160),(50,75,152),(51,67,153),(52,77,136),(53,78,155),(54,70,156)], [(2,11,20),(3,21,12),(5,14,23),(6,24,15),(8,17,26),(9,27,18),(28,37,46),(29,47,38),(31,40,49),(32,50,41),(34,43,52),(35,53,44),(55,64,73),(56,74,65),(58,67,76),(59,77,68),(61,70,79),(62,80,71),(82,100,91),(84,93,102),(85,103,94),(87,96,105),(88,106,97),(90,99,108),(109,118,127),(110,128,119),(112,121,130),(113,131,122),(115,124,133),(116,134,125),(137,146,155),(138,156,147),(140,149,158),(141,159,150),(143,152,161),(144,162,153)], [(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 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 9A ··· 9F 9G 9H 9I 9J 9K 9L 9M 9N 18A ··· 18F 18G 18H 18I 18J 18K 18L 18M 18N 27A ··· 27R 27S ··· 27AD 54A ··· 54R 54S ··· 54AD order 1 2 3 3 3 3 3 3 6 6 6 6 6 6 9 ··· 9 9 9 9 9 9 9 9 9 18 ··· 18 18 18 18 18 18 18 18 18 27 ··· 27 27 ··· 27 54 ··· 54 54 ··· 54 size 1 1 1 1 3 3 9 9 1 1 3 3 9 9 1 ··· 1 3 3 3 3 9 9 9 9 1 ··· 1 3 3 3 3 9 9 9 9 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C9 C18 C18 He3 3- 1+2 C2×He3 C2×3- 1+2 C9.6He3 C2×C9.6He3 kernel C2×C9.6He3 C9.6He3 C3×C54 C2×C27⋊C3 C2×C9○He3 C3×C27 C27⋊C3 C9○He3 C2×He3 C2×3- 1+2 He3 3- 1+2 C18 C18 C9 C9 C2 C1 # reps 1 1 2 4 2 2 4 2 6 12 6 12 2 4 2 4 18 18

Matrix representation of C2×C9.6He3 in GL3(𝔽109) generated by

 108 0 0 0 108 0 0 0 108
,
 27 0 0 0 27 0 0 0 27
,
 63 44 0 0 46 1 0 64 0
,
 1 0 0 63 45 0 108 0 63
,
 7 17 0 0 102 97 102 104 0
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[27,0,0,0,27,0,0,0,27],[63,0,0,44,46,64,0,1,0],[1,63,108,0,45,0,0,0,63],[7,0,102,17,102,104,0,97,0] >;

C2×C9.6He3 in GAP, Magma, Sage, TeX

C_2\times C_9._6{\rm He}_3
% in TeX

G:=Group("C2xC9.6He3");
// GroupNames label

G:=SmallGroup(486,80);
// by ID

G=gap.SmallGroup(486,80);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,735,208]);
// Polycyclic

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

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