direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C9.6He3, C18.5He3, He3.4C18, C18.23- 1+2, 3- 1+2.2C18, C27⋊C3⋊3C6, (C3×C54)⋊3C3, (C3×C27)⋊10C6, C9.5(C2×He3), C9○He3.4C6, (C2×He3).2C9, C32.4(C3×C18), C6.10(C32⋊C9), (C3×C18).25C32, C9.2(C2×3- 1+2), (C2×3- 1+2).2C9, (C2×C27⋊C3)⋊3C3, (C3×C6).4(C3×C9), (C3×C9).34(C3×C6), C3.10(C2×C32⋊C9), (C2×C9○He3).2C3, SmallGroup(486,80)
Series: Derived ►Chief ►Lower central ►Upper central
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
Export