direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C32⋊C27, C32⋊2C54, C18.2He3, C33.3C18, C18.63- 1+2, (C3×C6)⋊C27, (C3×C54)⋊1C3, (C3×C27)⋊8C6, C3.1(C3×C54), (C3×C18).2C9, (C3×C9).4C18, C6.1(C3×C27), C6.2(C27⋊C3), C9.2(C2×He3), (C32×C6).1C9, C6.2(C32⋊C9), (C32×C18).2C3, (C32×C9).14C6, (C3×C18).31C32, C32.20(C3×C18), C9.6(C2×3- 1+2), C3.2(C2×C27⋊C3), (C3×C6).15(C3×C9), (C3×C9).43(C3×C6), C3.2(C2×C32⋊C9), SmallGroup(486,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊C27
G = < a,b,c,d | a2=b3=c3=d27=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Subgroups: 126 in 70 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C9, C32, C32, C32, C18, C18, C18, C3×C6, C3×C6, C3×C6, C27, C3×C9, C3×C9, C3×C9, C33, C54, C3×C18, C3×C18, C3×C18, C32×C6, C3×C27, C32×C9, C3×C54, C32×C18, C32⋊C27, C2×C32⋊C27
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C27, C3×C9, He3, 3- 1+2, C54, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C3×C27, C27⋊C3, C2×C32⋊C9, C3×C54, C2×C27⋊C3, C32⋊C27, C2×C32⋊C27
►On 162 points
Generators in S162
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 161)(18 162)(19 136)(20 137)(21 138)(22 139)(23 140)(24 141)(25 142)(26 143)(27 144)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 109)
(2 43 119)(3 120 44)(5 46 122)(6 123 47)(8 49 125)(9 126 50)(11 52 128)(12 129 53)(14 28 131)(15 132 29)(17 31 134)(18 135 32)(20 34 110)(21 111 35)(23 37 113)(24 114 38)(26 40 116)(27 117 41)(55 137 88)(56 89 138)(58 140 91)(59 92 141)(61 143 94)(62 95 144)(64 146 97)(65 98 147)(67 149 100)(68 101 150)(70 152 103)(71 104 153)(73 155 106)(74 107 156)(76 158 82)(77 83 159)(79 161 85)(80 86 162)
(1 118 42)(2 119 43)(3 120 44)(4 121 45)(5 122 46)(6 123 47)(7 124 48)(8 125 49)(9 126 50)(10 127 51)(11 128 52)(12 129 53)(13 130 54)(14 131 28)(15 132 29)(16 133 30)(17 134 31)(18 135 32)(19 109 33)(20 110 34)(21 111 35)(22 112 36)(23 113 37)(24 114 38)(25 115 39)(26 116 40)(27 117 41)(55 88 137)(56 89 138)(57 90 139)(58 91 140)(59 92 141)(60 93 142)(61 94 143)(62 95 144)(63 96 145)(64 97 146)(65 98 147)(66 99 148)(67 100 149)(68 101 150)(69 102 151)(70 103 152)(71 104 153)(72 105 154)(73 106 155)(74 107 156)(75 108 157)(76 82 158)(77 83 159)(78 84 160)(79 85 161)(80 86 162)(81 87 136)
(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)| (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,161)(18,162)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,109), (2,43,119)(3,120,44)(5,46,122)(6,123,47)(8,49,125)(9,126,50)(11,52,128)(12,129,53)(14,28,131)(15,132,29)(17,31,134)(18,135,32)(20,34,110)(21,111,35)(23,37,113)(24,114,38)(26,40,116)(27,117,41)(55,137,88)(56,89,138)(58,140,91)(59,92,141)(61,143,94)(62,95,144)(64,146,97)(65,98,147)(67,149,100)(68,101,150)(70,152,103)(71,104,153)(73,155,106)(74,107,156)(76,158,82)(77,83,159)(79,161,85)(80,86,162), (1,118,42)(2,119,43)(3,120,44)(4,121,45)(5,122,46)(6,123,47)(7,124,48)(8,125,49)(9,126,50)(10,127,51)(11,128,52)(12,129,53)(13,130,54)(14,131,28)(15,132,29)(16,133,30)(17,134,31)(18,135,32)(19,109,33)(20,110,34)(21,111,35)(22,112,36)(23,113,37)(24,114,38)(25,115,39)(26,116,40)(27,117,41)(55,88,137)(56,89,138)(57,90,139)(58,91,140)(59,92,141)(60,93,142)(61,94,143)(62,95,144)(63,96,145)(64,97,146)(65,98,147)(66,99,148)(67,100,149)(68,101,150)(69,102,151)(70,103,152)(71,104,153)(72,105,154)(73,106,155)(74,107,156)(75,108,157)(76,82,158)(77,83,159)(78,84,160)(79,85,161)(80,86,162)(81,87,136), (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( (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,161)(18,162)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,109), (2,43,119)(3,120,44)(5,46,122)(6,123,47)(8,49,125)(9,126,50)(11,52,128)(12,129,53)(14,28,131)(15,132,29)(17,31,134)(18,135,32)(20,34,110)(21,111,35)(23,37,113)(24,114,38)(26,40,116)(27,117,41)(55,137,88)(56,89,138)(58,140,91)(59,92,141)(61,143,94)(62,95,144)(64,146,97)(65,98,147)(67,149,100)(68,101,150)(70,152,103)(71,104,153)(73,155,106)(74,107,156)(76,158,82)(77,83,159)(79,161,85)(80,86,162), (1,118,42)(2,119,43)(3,120,44)(4,121,45)(5,122,46)(6,123,47)(7,124,48)(8,125,49)(9,126,50)(10,127,51)(11,128,52)(12,129,53)(13,130,54)(14,131,28)(15,132,29)(16,133,30)(17,134,31)(18,135,32)(19,109,33)(20,110,34)(21,111,35)(22,112,36)(23,113,37)(24,114,38)(25,115,39)(26,116,40)(27,117,41)(55,88,137)(56,89,138)(57,90,139)(58,91,140)(59,92,141)(60,93,142)(61,94,143)(62,95,144)(63,96,145)(64,97,146)(65,98,147)(66,99,148)(67,100,149)(68,101,150)(69,102,151)(70,103,152)(71,104,153)(72,105,154)(73,106,155)(74,107,156)(75,108,157)(76,82,158)(77,83,159)(78,84,160)(79,85,161)(80,86,162)(81,87,136), (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([[(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,161),(18,162),(19,136),(20,137),(21,138),(22,139),(23,140),(24,141),(25,142),(26,143),(27,144),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,109)], [(2,43,119),(3,120,44),(5,46,122),(6,123,47),(8,49,125),(9,126,50),(11,52,128),(12,129,53),(14,28,131),(15,132,29),(17,31,134),(18,135,32),(20,34,110),(21,111,35),(23,37,113),(24,114,38),(26,40,116),(27,117,41),(55,137,88),(56,89,138),(58,140,91),(59,92,141),(61,143,94),(62,95,144),(64,146,97),(65,98,147),(67,149,100),(68,101,150),(70,152,103),(71,104,153),(73,155,106),(74,107,156),(76,158,82),(77,83,159),(79,161,85),(80,86,162)], [(1,118,42),(2,119,43),(3,120,44),(4,121,45),(5,122,46),(6,123,47),(7,124,48),(8,125,49),(9,126,50),(10,127,51),(11,128,52),(12,129,53),(13,130,54),(14,131,28),(15,132,29),(16,133,30),(17,134,31),(18,135,32),(19,109,33),(20,110,34),(21,111,35),(22,112,36),(23,113,37),(24,114,38),(25,115,39),(26,116,40),(27,117,41),(55,88,137),(56,89,138),(57,90,139),(58,91,140),(59,92,141),(60,93,142),(61,94,143),(62,95,144),(63,96,145),(64,97,146),(65,98,147),(66,99,148),(67,100,149),(68,101,150),(69,102,151),(70,103,152),(71,104,153),(72,105,154),(73,106,155),(74,107,156),(75,108,157),(76,82,158),(77,83,159),(78,84,160),(79,85,161),(80,86,162),(81,87,136)], [(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)]])
198 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 6A | ··· | 6H | 6I | ··· | 6N | 9A | ··· | 9R | 9S | ··· | 9AD | 18A | ··· | 18R | 18S | ··· | 18AD | 27A | ··· | 27BB | 54A | ··· | 54BB |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
198 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 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C54 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 | C27⋊C3 | C2×C27⋊C3 |
| kernel | C2×C32⋊C27 | C32⋊C27 | C3×C54 | C32×C18 | C3×C27 | C32×C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C3×C6 | C32 | C18 | C18 | C9 | C9 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 54 | 54 | 2 | 4 | 2 | 4 | 12 | 12 |
Matrix representation of C2×C32⋊C27 ►in GL4(𝔽109) generated by
| 108 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 45 | 0 | 0 | 0 |
| 0 | 1 | 72 | 70 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 45 |
| 15 | 0 | 0 | 0 |
| 0 | 66 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 46 | 4 | 43 |
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[45,0,0,0,0,1,0,0,0,72,45,0,0,70,0,63],[1,0,0,0,0,45,0,0,0,0,45,0,0,0,0,45],[15,0,0,0,0,66,0,46,0,0,0,4,0,0,1,43] >;
C2×C32⋊C27 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes C_{27} % in TeX
G:=Group("C2xC3^2:C27"); // GroupNames label
G:=SmallGroup(486,72);
// by ID
G=gap.SmallGroup(486,72);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^27=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations