direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C9.5He3, C18.4He3, He3.3C18, C18.13- 1+2, 3- 1+2.1C18, C27⋊C3⋊2C6, (C3×C54)⋊2C3, (C3×C27)⋊9C6, C9.4(C2×He3), C9○He3.3C6, (C2×He3).1C9, C6.9(C32⋊C9), C32.3(C3×C18), (C3×C18).24C32, C9.1(C2×3- 1+2), (C2×3- 1+2).1C9, (C2×C27⋊C3)⋊2C3, (C3×C6).3(C3×C9), (C3×C9).33(C3×C6), C3.9(C2×C32⋊C9), (C2×C9○He3).1C3, SmallGroup(486,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9.5He3
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=b3c, ece-1=b3cd-1, ede-1=b6d >
Smallest permutation representation of C2×C9.5He3
►On 162 points
Generators in S162
(1 140)(2 141)(3 142)(4 143)(5 144)(6 145)(7 146)(8 147)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 160)(22 161)(23 162)(24 136)(25 137)(26 138)(27 139)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)
(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 44 114)(2 36 133)(3 37 116)(4 47 117)(5 39 109)(6 40 119)(7 50 120)(8 42 112)(9 43 122)(10 53 123)(11 45 115)(12 46 125)(13 29 126)(14 48 118)(15 49 128)(16 32 129)(17 51 121)(18 52 131)(19 35 132)(20 54 124)(21 28 134)(22 38 135)(23 30 127)(24 31 110)(25 41 111)(26 33 130)(27 34 113)(55 146 95)(56 156 96)(57 148 88)(58 149 98)(59 159 99)(60 151 91)(61 152 101)(62 162 102)(63 154 94)(64 155 104)(65 138 105)(66 157 97)(67 158 107)(68 141 108)(69 160 100)(70 161 83)(71 144 84)(72 136 103)(73 137 86)(74 147 87)(75 139 106)(76 140 89)(77 150 90)(78 142 82)(79 143 92)(80 153 93)(81 145 85)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)(55 64 73)(57 75 66)(58 67 76)(60 78 69)(61 70 79)(63 81 72)(83 101 92)(84 93 102)(86 104 95)(87 96 105)(89 107 98)(90 99 108)(110 128 119)(111 120 129)(113 131 122)(114 123 132)(116 134 125)(117 126 135)(136 145 154)(138 156 147)(139 148 157)(141 159 150)(142 151 160)(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)| (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,136)(25,137)(26,138)(27,139)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119), (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), (1,44,114)(2,36,133)(3,37,116)(4,47,117)(5,39,109)(6,40,119)(7,50,120)(8,42,112)(9,43,122)(10,53,123)(11,45,115)(12,46,125)(13,29,126)(14,48,118)(15,49,128)(16,32,129)(17,51,121)(18,52,131)(19,35,132)(20,54,124)(21,28,134)(22,38,135)(23,30,127)(24,31,110)(25,41,111)(26,33,130)(27,34,113)(55,146,95)(56,156,96)(57,148,88)(58,149,98)(59,159,99)(60,151,91)(61,152,101)(62,162,102)(63,154,94)(64,155,104)(65,138,105)(66,157,97)(67,158,107)(68,141,108)(69,160,100)(70,161,83)(71,144,84)(72,136,103)(73,137,86)(74,147,87)(75,139,106)(76,140,89)(77,150,90)(78,142,82)(79,143,92)(80,153,93)(81,145,85), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108)(110,128,119)(111,120,129)(113,131,122)(114,123,132)(116,134,125)(117,126,135)(136,145,154)(138,156,147)(139,148,157)(141,159,150)(142,151,160)(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( (1,140)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,160)(22,161)(23,162)(24,136)(25,137)(26,138)(27,139)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119), (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), (1,44,114)(2,36,133)(3,37,116)(4,47,117)(5,39,109)(6,40,119)(7,50,120)(8,42,112)(9,43,122)(10,53,123)(11,45,115)(12,46,125)(13,29,126)(14,48,118)(15,49,128)(16,32,129)(17,51,121)(18,52,131)(19,35,132)(20,54,124)(21,28,134)(22,38,135)(23,30,127)(24,31,110)(25,41,111)(26,33,130)(27,34,113)(55,146,95)(56,156,96)(57,148,88)(58,149,98)(59,159,99)(60,151,91)(61,152,101)(62,162,102)(63,154,94)(64,155,104)(65,138,105)(66,157,97)(67,158,107)(68,141,108)(69,160,100)(70,161,83)(71,144,84)(72,136,103)(73,137,86)(74,147,87)(75,139,106)(76,140,89)(77,150,90)(78,142,82)(79,143,92)(80,153,93)(81,145,85), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108)(110,128,119)(111,120,129)(113,131,122)(114,123,132)(116,134,125)(117,126,135)(136,145,154)(138,156,147)(139,148,157)(141,159,150)(142,151,160)(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([[(1,140),(2,141),(3,142),(4,143),(5,144),(6,145),(7,146),(8,147),(9,148),(10,149),(11,150),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,160),(22,161),(23,162),(24,136),(25,137),(26,138),(27,139),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119)], [(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)], [(1,44,114),(2,36,133),(3,37,116),(4,47,117),(5,39,109),(6,40,119),(7,50,120),(8,42,112),(9,43,122),(10,53,123),(11,45,115),(12,46,125),(13,29,126),(14,48,118),(15,49,128),(16,32,129),(17,51,121),(18,52,131),(19,35,132),(20,54,124),(21,28,134),(22,38,135),(23,30,127),(24,31,110),(25,41,111),(26,33,130),(27,34,113),(55,146,95),(56,156,96),(57,148,88),(58,149,98),(59,159,99),(60,151,91),(61,152,101),(62,162,102),(63,154,94),(64,155,104),(65,138,105),(66,157,97),(67,158,107),(68,141,108),(69,160,100),(70,161,83),(71,144,84),(72,136,103),(73,137,86),(74,147,87),(75,139,106),(76,140,89),(77,150,90),(78,142,82),(79,143,92),(80,153,93),(81,145,85)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54),(55,64,73),(57,75,66),(58,67,76),(60,78,69),(61,70,79),(63,81,72),(83,101,92),(84,93,102),(86,104,95),(87,96,105),(89,107,98),(90,99,108),(110,128,119),(111,120,129),(113,131,122),(114,123,132),(116,134,125),(117,126,135),(136,145,154),(138,156,147),(139,148,157),(141,159,150),(142,151,160),(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 | C9.5He3 | C2×C9.5He3 |
| kernel | C2×C9.5He3 | C9.5He3 | 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×C9.5He3 ►in GL4(𝔽109) generated by
| 108 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 78 | 0 | 0 |
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,45,0,0,0,0,63],[1,0,0,0,0,0,0,78,0,22,0,0,0,0,22,0] >;
C2×C9.5He3 in GAP, Magma, Sage, TeX
C_2\times C_9._5{\rm He}_3 % in TeX
G:=Group("C2xC9.5He3"); // GroupNames label
G:=SmallGroup(486,79);
// by ID
G=gap.SmallGroup(486,79);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,735,118]);
// 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^3*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations
Export