direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C92⋊7C3, C92⋊20C6, C18⋊13- 1+2, C9⋊C9⋊8C6, (C9×C18)⋊7C3, C32⋊C9.19C6, C6.8(C9○He3), C33.10(C3×C6), (C3×C18).9C32, (C3×C6).26C33, (C32×C6).9C32, C9⋊4(C2×3- 1+2), C32.30(C32×C6), C6.8(C3×3- 1+2), C3.8(C6×3- 1+2), (C6×3- 1+2).5C3, (C3×3- 1+2).8C6, (C2×C9⋊C9)⋊5C3, (C3×C9).26(C3×C6), C3.8(C2×C9○He3), (C2×C32⋊C9).10C3, SmallGroup(486,202)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C92⋊7C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc6, dcd-1=b6c >
Subgroups: 216 in 120 conjugacy classes, 78 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C32, C18, C18, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C92, C32⋊C9, C9⋊C9, C3×3- 1+2, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C6×3- 1+2, C92⋊7C3, C2×C92⋊7C3
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C9○He3, C6×3- 1+2, C2×C9○He3, C92⋊7C3, C2×C92⋊7C3
►On 162 points
Generators in S162
(1 143)(2 144)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 88)(29 89)(30 90)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 115)(56 116)(57 117)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)
(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)
(1 116 89 80 25 41 53 12 71)(2 117 90 81 26 42 54 13 72)(3 109 82 73 27 43 46 14 64)(4 110 83 74 19 44 47 15 65)(5 111 84 75 20 45 48 16 66)(6 112 85 76 21 37 49 17 67)(7 113 86 77 22 38 50 18 68)(8 114 87 78 23 39 51 10 69)(9 115 88 79 24 40 52 11 70)(28 160 105 121 133 92 151 142 55)(29 161 106 122 134 93 152 143 56)(30 162 107 123 135 94 153 144 57)(31 154 108 124 127 95 145 136 58)(32 155 100 125 128 96 146 137 59)(33 156 101 126 129 97 147 138 60)(34 157 102 118 130 98 148 139 61)(35 158 103 119 131 99 149 140 62)(36 159 104 120 132 91 150 141 63)
(2 81 54)(3 46 73)(5 75 48)(6 49 76)(8 78 51)(9 52 79)(10 117 20)(11 27 112)(12 15 18)(13 111 23)(14 21 115)(16 114 26)(17 24 109)(19 22 25)(28 148 124)(29 35 32)(30 120 147)(31 151 118)(33 123 150)(34 145 121)(36 126 153)(37 82 70)(38 44 41)(39 66 90)(40 85 64)(42 69 84)(43 88 67)(45 72 87)(55 95 102)(56 59 62)(57 101 91)(58 98 105)(60 104 94)(61 92 108)(63 107 97)(65 71 68)(83 89 86)(93 96 99)(100 103 106)(110 113 116)(119 125 122)(127 154 136)(129 138 156)(130 157 139)(132 141 159)(133 160 142)(135 144 162)(146 152 149)
G:=sub<Sym(162)| (1,143)(2,144)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,88)(29,89)(30,90)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,115)(56,116)(57,117)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162), (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), (1,116,89,80,25,41,53,12,71)(2,117,90,81,26,42,54,13,72)(3,109,82,73,27,43,46,14,64)(4,110,83,74,19,44,47,15,65)(5,111,84,75,20,45,48,16,66)(6,112,85,76,21,37,49,17,67)(7,113,86,77,22,38,50,18,68)(8,114,87,78,23,39,51,10,69)(9,115,88,79,24,40,52,11,70)(28,160,105,121,133,92,151,142,55)(29,161,106,122,134,93,152,143,56)(30,162,107,123,135,94,153,144,57)(31,154,108,124,127,95,145,136,58)(32,155,100,125,128,96,146,137,59)(33,156,101,126,129,97,147,138,60)(34,157,102,118,130,98,148,139,61)(35,158,103,119,131,99,149,140,62)(36,159,104,120,132,91,150,141,63), (2,81,54)(3,46,73)(5,75,48)(6,49,76)(8,78,51)(9,52,79)(10,117,20)(11,27,112)(12,15,18)(13,111,23)(14,21,115)(16,114,26)(17,24,109)(19,22,25)(28,148,124)(29,35,32)(30,120,147)(31,151,118)(33,123,150)(34,145,121)(36,126,153)(37,82,70)(38,44,41)(39,66,90)(40,85,64)(42,69,84)(43,88,67)(45,72,87)(55,95,102)(56,59,62)(57,101,91)(58,98,105)(60,104,94)(61,92,108)(63,107,97)(65,71,68)(83,89,86)(93,96,99)(100,103,106)(110,113,116)(119,125,122)(127,154,136)(129,138,156)(130,157,139)(132,141,159)(133,160,142)(135,144,162)(146,152,149)>;
G:=Group( (1,143)(2,144)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,88)(29,89)(30,90)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,115)(56,116)(57,117)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162), (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), (1,116,89,80,25,41,53,12,71)(2,117,90,81,26,42,54,13,72)(3,109,82,73,27,43,46,14,64)(4,110,83,74,19,44,47,15,65)(5,111,84,75,20,45,48,16,66)(6,112,85,76,21,37,49,17,67)(7,113,86,77,22,38,50,18,68)(8,114,87,78,23,39,51,10,69)(9,115,88,79,24,40,52,11,70)(28,160,105,121,133,92,151,142,55)(29,161,106,122,134,93,152,143,56)(30,162,107,123,135,94,153,144,57)(31,154,108,124,127,95,145,136,58)(32,155,100,125,128,96,146,137,59)(33,156,101,126,129,97,147,138,60)(34,157,102,118,130,98,148,139,61)(35,158,103,119,131,99,149,140,62)(36,159,104,120,132,91,150,141,63), (2,81,54)(3,46,73)(5,75,48)(6,49,76)(8,78,51)(9,52,79)(10,117,20)(11,27,112)(12,15,18)(13,111,23)(14,21,115)(16,114,26)(17,24,109)(19,22,25)(28,148,124)(29,35,32)(30,120,147)(31,151,118)(33,123,150)(34,145,121)(36,126,153)(37,82,70)(38,44,41)(39,66,90)(40,85,64)(42,69,84)(43,88,67)(45,72,87)(55,95,102)(56,59,62)(57,101,91)(58,98,105)(60,104,94)(61,92,108)(63,107,97)(65,71,68)(83,89,86)(93,96,99)(100,103,106)(110,113,116)(119,125,122)(127,154,136)(129,138,156)(130,157,139)(132,141,159)(133,160,142)(135,144,162)(146,152,149) );
G=PermutationGroup([[(1,143),(2,144),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,88),(29,89),(30,90),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,115),(56,116),(57,117),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162)], [(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)], [(1,116,89,80,25,41,53,12,71),(2,117,90,81,26,42,54,13,72),(3,109,82,73,27,43,46,14,64),(4,110,83,74,19,44,47,15,65),(5,111,84,75,20,45,48,16,66),(6,112,85,76,21,37,49,17,67),(7,113,86,77,22,38,50,18,68),(8,114,87,78,23,39,51,10,69),(9,115,88,79,24,40,52,11,70),(28,160,105,121,133,92,151,142,55),(29,161,106,122,134,93,152,143,56),(30,162,107,123,135,94,153,144,57),(31,154,108,124,127,95,145,136,58),(32,155,100,125,128,96,146,137,59),(33,156,101,126,129,97,147,138,60),(34,157,102,118,130,98,148,139,61),(35,158,103,119,131,99,149,140,62),(36,159,104,120,132,91,150,141,63)], [(2,81,54),(3,46,73),(5,75,48),(6,49,76),(8,78,51),(9,52,79),(10,117,20),(11,27,112),(12,15,18),(13,111,23),(14,21,115),(16,114,26),(17,24,109),(19,22,25),(28,148,124),(29,35,32),(30,120,147),(31,151,118),(33,123,150),(34,145,121),(36,126,153),(37,82,70),(38,44,41),(39,66,90),(40,85,64),(42,69,84),(43,88,67),(45,72,87),(55,95,102),(56,59,62),(57,101,91),(58,98,105),(60,104,94),(61,92,108),(63,107,97),(65,71,68),(83,89,86),(93,96,99),(100,103,106),(110,113,116),(119,125,122),(127,154,136),(129,138,156),(130,157,139),(132,141,159),(133,160,142),(135,144,162),(146,152,149)]])
102 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | 3J | 6A | ··· | 6H | 6I | 6J | 9A | ··· | 9X | 9Y | ··· | 9AN | 18A | ··· | 18X | 18Y | ··· | 18AN |
| order | 1 | 2 | 3 | ··· | 3 | 3 | 3 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 9 | 9 | 1 | ··· | 1 | 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 | 3- 1+2 | C2×3- 1+2 | C9○He3 | C2×C9○He3 |
| kernel | C2×C92⋊7C3 | C92⋊7C3 | C9×C18 | C2×C32⋊C9 | C2×C9⋊C9 | C6×3- 1+2 | C92 | C32⋊C9 | C9⋊C9 | C3×3- 1+2 | C18 | C9 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 16 | 4 | 2 | 4 | 16 | 4 | 12 | 12 | 12 | 12 |
Matrix representation of C2×C92⋊7C3 ►in GL6(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,11,0,0,0,0,0,0,11,0],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7] >;
C2×C92⋊7C3 in GAP, Magma, Sage, TeX
C_2\times C_9^2\rtimes_7C_3
% in TeX
G:=Group("C2xC9^2:7C3"); // GroupNames label
G:=SmallGroup(486,202);
// by ID
G=gap.SmallGroup(486,202);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,500,2169,93]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^6,d*c*d^-1=b^6*c>;
// generators/relations