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## G = C6×C27⋊C3order 486 = 2·35

### Direct product of C6 and C27⋊C3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×C27⋊C3
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C3×C27⋊C3 — C6×C27⋊C3
 Lower central C1 — C3 — C6×C27⋊C3
 Upper central C1 — C3×C18 — C6×C27⋊C3

Generators and relations for C6×C27⋊C3
G = < a,b,c | a6=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >

Subgroups: 144 in 120 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C32, C32, C32, C18, C18, C3×C6, C3×C6, C3×C6, C27, C3×C9, C3×C9, C33, C54, C3×C18, C3×C18, C32×C6, C3×C27, C27⋊C3, C32×C9, C3×C54, C2×C27⋊C3, C32×C18, C3×C27⋊C3, C6×C27⋊C3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C27⋊C3, C32×C9, C2×C27⋊C3, C32×C18, C3×C27⋊C3, C6×C27⋊C3

Smallest permutation representation of C6×C27⋊C3
On 162 points
Generators in S162
(1 129 143 84 66 28)(2 130 144 85 67 29)(3 131 145 86 68 30)(4 132 146 87 69 31)(5 133 147 88 70 32)(6 134 148 89 71 33)(7 135 149 90 72 34)(8 109 150 91 73 35)(9 110 151 92 74 36)(10 111 152 93 75 37)(11 112 153 94 76 38)(12 113 154 95 77 39)(13 114 155 96 78 40)(14 115 156 97 79 41)(15 116 157 98 80 42)(16 117 158 99 81 43)(17 118 159 100 55 44)(18 119 160 101 56 45)(19 120 161 102 57 46)(20 121 162 103 58 47)(21 122 136 104 59 48)(22 123 137 105 60 49)(23 124 138 106 61 50)(24 125 139 107 62 51)(25 126 140 108 63 52)(26 127 141 82 64 53)(27 128 142 83 65 54)
(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 57 152)(2 76 162)(3 68 145)(4 60 155)(5 79 138)(6 71 148)(7 63 158)(8 55 141)(9 74 151)(10 66 161)(11 58 144)(12 77 154)(13 69 137)(14 61 147)(15 80 157)(16 72 140)(17 64 150)(18 56 160)(19 75 143)(20 67 153)(21 59 136)(22 78 146)(23 70 156)(24 62 139)(25 81 149)(26 73 159)(27 65 142)(28 102 111)(29 94 121)(30 86 131)(31 105 114)(32 97 124)(33 89 134)(34 108 117)(35 100 127)(36 92 110)(37 84 120)(38 103 130)(39 95 113)(40 87 123)(41 106 133)(42 98 116)(43 90 126)(44 82 109)(45 101 119)(46 93 129)(47 85 112)(48 104 122)(49 96 132)(50 88 115)(51 107 125)(52 99 135)(53 91 118)(54 83 128)

G:=sub<Sym(162)| (1,129,143,84,66,28)(2,130,144,85,67,29)(3,131,145,86,68,30)(4,132,146,87,69,31)(5,133,147,88,70,32)(6,134,148,89,71,33)(7,135,149,90,72,34)(8,109,150,91,73,35)(9,110,151,92,74,36)(10,111,152,93,75,37)(11,112,153,94,76,38)(12,113,154,95,77,39)(13,114,155,96,78,40)(14,115,156,97,79,41)(15,116,157,98,80,42)(16,117,158,99,81,43)(17,118,159,100,55,44)(18,119,160,101,56,45)(19,120,161,102,57,46)(20,121,162,103,58,47)(21,122,136,104,59,48)(22,123,137,105,60,49)(23,124,138,106,61,50)(24,125,139,107,62,51)(25,126,140,108,63,52)(26,127,141,82,64,53)(27,128,142,83,65,54), (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), (1,57,152)(2,76,162)(3,68,145)(4,60,155)(5,79,138)(6,71,148)(7,63,158)(8,55,141)(9,74,151)(10,66,161)(11,58,144)(12,77,154)(13,69,137)(14,61,147)(15,80,157)(16,72,140)(17,64,150)(18,56,160)(19,75,143)(20,67,153)(21,59,136)(22,78,146)(23,70,156)(24,62,139)(25,81,149)(26,73,159)(27,65,142)(28,102,111)(29,94,121)(30,86,131)(31,105,114)(32,97,124)(33,89,134)(34,108,117)(35,100,127)(36,92,110)(37,84,120)(38,103,130)(39,95,113)(40,87,123)(41,106,133)(42,98,116)(43,90,126)(44,82,109)(45,101,119)(46,93,129)(47,85,112)(48,104,122)(49,96,132)(50,88,115)(51,107,125)(52,99,135)(53,91,118)(54,83,128)>;

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

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

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 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C9 C18 C18 C27⋊C3 C2×C27⋊C3 kernel C6×C27⋊C3 C3×C27⋊C3 C3×C54 C2×C27⋊C3 C32×C18 C3×C27 C27⋊C3 C32×C9 C3×C18 C32×C6 C3×C9 C33 C6 C3 # reps 1 1 6 18 2 6 18 2 48 6 48 6 18 18

Matrix representation of C6×C27⋊C3 in GL4(𝔽109) generated by

 63 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 63 0 0 0 0 58 21 2 0 0 0 63 0 70 43 51
,
 1 0 0 0 0 63 64 95 0 0 45 0 0 0 0 1
G:=sub<GL(4,GF(109))| [63,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[63,0,0,0,0,58,0,70,0,21,0,43,0,2,63,51],[1,0,0,0,0,63,0,0,0,64,45,0,0,95,0,1] >;

C6×C27⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_{27}\rtimes C_3
% in TeX

G:=Group("C6xC27:C3");
// GroupNames label

G:=SmallGroup(486,208);
// by ID

G=gap.SmallGroup(486,208);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,1520,118]);
// Polycyclic

G:=Group<a,b,c|a^6=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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