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## G = C2×C27○He3order 486 = 2·35

### Direct product of C2 and C27○He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C27○He3
 Chief series C1 — C3 — C9 — C3×C9 — C9○He3 — C27○He3 — C2×C27○He3
 Lower central C1 — C3 — C2×C27○He3
 Upper central C1 — C54 — C2×C27○He3

Generators and relations for C2×C27○He3
G = < a,b,c,d,e | a2=b27=c3=e3=1, d1=b18, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b9c, de=ed >

Subgroups: 126 in 110 conjugacy classes, 102 normal (14 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C18, C18, C3×C6, C27, C27, C3×C9, He3, 3- 1+2, C54, C54, C3×C18, C2×He3, C2×3- 1+2, C3×C27, C27⋊C3, C9○He3, C3×C54, C2×C27⋊C3, C2×C9○He3, C27○He3, C2×C27○He3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C32×C18, C27○He3, C2×C27○He3

Smallest permutation representation of C2×C27○He3
On 162 points
Generators in S162
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 55)(27 56)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 162)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 109)(99 110)(100 111)(101 112)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)
(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)
(1 159 82)(2 160 83)(3 161 84)(4 162 85)(5 136 86)(6 137 87)(7 138 88)(8 139 89)(9 140 90)(10 141 91)(11 142 92)(12 143 93)(13 144 94)(14 145 95)(15 146 96)(16 147 97)(17 148 98)(18 149 99)(19 150 100)(20 151 101)(21 152 102)(22 153 103)(23 154 104)(24 155 105)(25 156 106)(26 157 107)(27 158 108)(28 133 70)(29 134 71)(30 135 72)(31 109 73)(32 110 74)(33 111 75)(34 112 76)(35 113 77)(36 114 78)(37 115 79)(38 116 80)(39 117 81)(40 118 55)(41 119 56)(42 120 57)(43 121 58)(44 122 59)(45 123 60)(46 124 61)(47 125 62)(48 126 63)(49 127 64)(50 128 65)(51 129 66)(52 130 67)(53 131 68)(54 132 69)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(55 73 64)(56 74 65)(57 75 66)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)(82 100 91)(83 101 92)(84 102 93)(85 103 94)(86 104 95)(87 105 96)(88 106 97)(89 107 98)(90 108 99)(109 127 118)(110 128 119)(111 129 120)(112 130 121)(113 131 122)(114 132 123)(115 133 124)(116 134 125)(117 135 126)(136 154 145)(137 155 146)(138 156 147)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)
(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(82 91 100)(83 92 101)(84 93 102)(85 94 103)(86 95 104)(87 96 105)(88 97 106)(89 98 107)(90 99 108)(109 118 127)(110 119 128)(111 120 129)(112 121 130)(113 122 131)(114 123 132)(115 124 133)(116 125 134)(117 126 135)(136 154 145)(137 155 146)(138 156 147)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)

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

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

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

198 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 6A 6B 6C ··· 6J 9A ··· 9F 9G ··· 9V 18A ··· 18F 18G ··· 18V 27A ··· 27R 27S ··· 27BN 54A ··· 54R 54S ··· 54BN order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 54 ··· 54 size 1 1 1 1 3 ··· 3 1 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 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○He3 C2×C27○He3 kernel C2×C27○He3 C27○He3 C3×C54 C2×C27⋊C3 C2×C9○He3 C3×C27 C27⋊C3 C9○He3 C2×He3 C2×3- 1+2 He3 3- 1+2 C2 C1 # reps 1 1 8 16 2 8 16 2 6 48 6 48 18 18

Matrix representation of C2×C27○He3 in GL3(𝔽109) generated by

 108 0 0 0 108 0 0 0 108
,
 73 0 0 0 73 0 0 0 73
,
 63 0 0 0 0 1 44 64 46
,
 63 0 0 0 63 0 0 0 63
,
 1 45 64 0 45 0 0 0 63
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[73,0,0,0,73,0,0,0,73],[63,0,44,0,0,64,0,1,46],[63,0,0,0,63,0,0,0,63],[1,0,0,45,45,0,64,0,63] >;

C2×C27○He3 in GAP, Magma, Sage, TeX

C_2\times C_{27}\circ {\rm He}_3
% in TeX

G:=Group("C2xC27oHe3");
// GroupNames label

G:=SmallGroup(486,209);
// by ID

G=gap.SmallGroup(486,209);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,735,118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^27=c^3=e^3=1,d^1=b^18,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,c*d=d*c,e*c*e^-1=b^9*c,d*e=e*d>;
// generators/relations

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