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G = C3×C6×He3order 486 = 2·35

Direct product of C3×C6 and He3

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

Aliases: C3×C6×He3, C3413C6, C6.1C34, (C3×C6)⋊C33, (C33×C6)⋊3C3, C3312(C3×C6), C3.1(C33×C6), (C32×C6)⋊5C32, C322(C32×C6), SmallGroup(486,251)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×C6×He3
 Chief series C1 — C3 — C32 — C33 — C34 — C32×He3 — C3×C6×He3
 Lower central C1 — C3 — C3×C6×He3
 Upper central C1 — C32×C6 — C3×C6×He3

Generators and relations for C3×C6×He3
G = < a,b,c,d,e | a3=b6=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 1764 in 900 conjugacy classes, 468 normal (8 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C32, C32, C3×C6, C3×C6, He3, C33, C33, C33, C2×He3, C32×C6, C32×C6, C32×C6, C3×He3, C34, C6×He3, C33×C6, C32×He3, C3×C6×He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C34, C6×He3, C33×C6, C32×He3, C3×C6×He3

Smallest permutation representation of C3×C6×He3
On 162 points
Generators in S162
(1 105 135)(2 106 136)(3 107 137)(4 108 138)(5 103 133)(6 104 134)(7 158 16)(8 159 17)(9 160 18)(10 161 13)(11 162 14)(12 157 15)(19 59 45)(20 60 46)(21 55 47)(22 56 48)(23 57 43)(24 58 44)(25 67 120)(26 68 115)(27 69 116)(28 70 117)(29 71 118)(30 72 119)(31 73 65)(32 74 66)(33 75 61)(34 76 62)(35 77 63)(36 78 64)(37 130 84)(38 131 79)(39 132 80)(40 127 81)(41 128 82)(42 129 83)(49 124 85)(50 125 86)(51 126 87)(52 121 88)(53 122 89)(54 123 90)(91 142 102)(92 143 97)(93 144 98)(94 139 99)(95 140 100)(96 141 101)(109 149 156)(110 150 151)(111 145 152)(112 146 153)(113 147 154)(114 148 155)
(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 54 115)(2 49 116)(3 50 117)(4 51 118)(5 52 119)(6 53 120)(7 75 95)(8 76 96)(9 77 91)(10 78 92)(11 73 93)(12 74 94)(13 36 97)(14 31 98)(15 32 99)(16 33 100)(17 34 101)(18 35 102)(19 37 111)(20 38 112)(21 39 113)(22 40 114)(23 41 109)(24 42 110)(25 104 122)(26 105 123)(27 106 124)(28 107 125)(29 108 126)(30 103 121)(43 82 156)(44 83 151)(45 84 152)(46 79 153)(47 80 154)(48 81 155)(55 132 147)(56 127 148)(57 128 149)(58 129 150)(59 130 145)(60 131 146)(61 140 158)(62 141 159)(63 142 160)(64 143 161)(65 144 162)(66 139 157)(67 134 89)(68 135 90)(69 136 85)(70 137 86)(71 138 87)(72 133 88)
(1 48 96)(2 43 91)(3 44 92)(4 45 93)(5 46 94)(6 47 95)(7 53 80)(8 54 81)(9 49 82)(10 50 83)(11 51 84)(12 52 79)(13 86 129)(14 87 130)(15 88 131)(16 89 132)(17 90 127)(18 85 128)(19 144 108)(20 139 103)(21 140 104)(22 141 105)(23 142 106)(24 143 107)(25 113 61)(26 114 62)(27 109 63)(28 110 64)(29 111 65)(30 112 66)(31 71 145)(32 72 146)(33 67 147)(34 68 148)(35 69 149)(36 70 150)(37 162 126)(38 157 121)(39 158 122)(40 159 123)(41 160 124)(42 161 125)(55 100 134)(56 101 135)(57 102 136)(58 97 137)(59 98 138)(60 99 133)(73 118 152)(74 119 153)(75 120 154)(76 115 155)(77 116 156)(78 117 151)
(1 20 97)(2 21 98)(3 22 99)(4 23 100)(5 24 101)(6 19 102)(7 37 85)(8 38 86)(9 39 87)(10 40 88)(11 41 89)(12 42 90)(13 81 121)(14 82 122)(15 83 123)(16 84 124)(17 79 125)(18 80 126)(25 71 116)(26 72 117)(27 67 118)(28 68 119)(29 69 120)(30 70 115)(31 77 61)(32 78 62)(33 73 63)(34 74 64)(35 75 65)(36 76 66)(43 140 138)(44 141 133)(45 142 134)(46 143 135)(47 144 136)(48 139 137)(49 158 130)(50 159 131)(51 160 132)(52 161 127)(53 162 128)(54 157 129)(55 93 106)(56 94 107)(57 95 108)(58 96 103)(59 91 104)(60 92 105)(109 147 152)(110 148 153)(111 149 154)(112 150 155)(113 145 156)(114 146 151)

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

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

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

198 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3CT 6A ··· 6Z 6AA ··· 6CT order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 kernel C3×C6×He3 C32×He3 C6×He3 C33×C6 C3×He3 C34 C3×C6 C32 # reps 1 1 72 8 72 8 18 18

Matrix representation of C3×C6×He3 in GL5(𝔽7)

 4 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 3 0 0 0 0 0 5 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 5 5 5 0 0 0 0 1 0 0 1 5 2
,
 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 4 0 0 0 0 0 1 2 1 0 0 0 2 0 0 0 0 0 4

G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[3,0,0,0,0,0,5,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,1,0,0,5,0,5,0,0,5,1,2],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,2,2,0,0,0,1,0,4] >;

C3×C6×He3 in GAP, Magma, Sage, TeX

C_3\times C_6\times {\rm He}_3
% in TeX

G:=Group("C3xC6xHe3");
// GroupNames label

G:=SmallGroup(486,251);
// by ID

G=gap.SmallGroup(486,251);
# by ID

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

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

׿
×
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