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## G = C2×C32⋊C27order 486 = 2·35

### Direct product of C2 and C32⋊C27

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C32⋊C27
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32⋊C27 — C2×C32⋊C27
 Lower central C1 — C3 — C2×C32⋊C27
 Upper central C1 — C3×C18 — C2×C32⋊C27

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

Smallest permutation representation of 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

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