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## G = C2×C9⋊He3order 486 = 2·35

### Direct product of C2 and C9⋊He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C9⋊He3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9⋊He3 — C2×C9⋊He3
 Lower central C1 — C32 — C2×C9⋊He3
 Upper central C1 — C3×C6 — C2×C9⋊He3

Generators and relations for C2×C9⋊He3
G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 378 in 166 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C32, C32, C32, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, C3×C18, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C32×C6, C32×C6, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C2×C32⋊C9, C32×C18, C6×He3, C6×3- 1+2, C9⋊He3, C2×C9⋊He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, 3- 1+2, C33, C2×He3, C2×3- 1+2, C32×C6, C3×He3, C3×3- 1+2, C9○He3, C6×He3, C6×3- 1+2, C2×C9○He3, C9⋊He3, C2×C9⋊He3

Smallest permutation representation of C2×C9⋊He3
On 162 points
Generators in S162
(1 105)(2 106)(3 107)(4 108)(5 100)(6 101)(7 102)(8 103)(9 104)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 82)(18 83)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 91)(27 92)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)
(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)
(10 159 26)(11 160 27)(12 161 19)(13 162 20)(14 154 21)(15 155 22)(16 156 23)(17 157 24)(18 158 25)(46 71 58)(47 72 59)(48 64 60)(49 65 61)(50 66 62)(51 67 63)(52 68 55)(53 69 56)(54 70 57)(73 95 88)(74 96 89)(75 97 90)(76 98 82)(77 99 83)(78 91 84)(79 92 85)(80 93 86)(81 94 87)(127 152 139)(128 153 140)(129 145 141)(130 146 142)(131 147 143)(132 148 144)(133 149 136)(134 150 137)(135 151 138)
(1 41 28)(2 42 29)(3 43 30)(4 44 31)(5 45 32)(6 37 33)(7 38 34)(8 39 35)(9 40 36)(10 26 159)(11 27 160)(12 19 161)(13 20 162)(14 21 154)(15 22 155)(16 23 156)(17 24 157)(18 25 158)(46 71 58)(47 72 59)(48 64 60)(49 65 61)(50 66 62)(51 67 63)(52 68 55)(53 69 56)(54 70 57)(73 88 95)(74 89 96)(75 90 97)(76 82 98)(77 83 99)(78 84 91)(79 85 92)(80 86 93)(81 87 94)(100 126 113)(101 118 114)(102 119 115)(103 120 116)(104 121 117)(105 122 109)(106 123 110)(107 124 111)(108 125 112)(127 152 139)(128 153 140)(129 145 141)(130 146 142)(131 147 143)(132 148 144)(133 149 136)(134 150 137)(135 151 138)
(1 88 52)(2 83 50)(3 87 48)(4 82 46)(5 86 53)(6 90 51)(7 85 49)(8 89 47)(9 84 54)(10 135 104)(11 130 102)(12 134 100)(13 129 107)(14 133 105)(15 128 103)(16 132 101)(17 127 108)(18 131 106)(19 150 126)(20 145 124)(21 149 122)(22 153 120)(23 148 118)(24 152 125)(25 147 123)(26 151 121)(27 146 119)(28 73 55)(29 77 62)(30 81 60)(31 76 58)(32 80 56)(33 75 63)(34 79 61)(35 74 59)(36 78 57)(37 97 67)(38 92 65)(39 96 72)(40 91 70)(41 95 68)(42 99 66)(43 94 64)(44 98 71)(45 93 69)(109 154 136)(110 158 143)(111 162 141)(112 157 139)(113 161 137)(114 156 144)(115 160 142)(116 155 140)(117 159 138)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 3O ··· 3T 6A ··· 6H 6I ··· 6N 6O ··· 6T 9A ··· 9R 9S ··· 9AD 18A ··· 18R 18S ··· 18AD order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 He3 3- 1+2 C2×He3 C2×3- 1+2 C9○He3 C2×C9○He3 kernel C2×C9⋊He3 C9⋊He3 C2×C32⋊C9 C32×C18 C6×He3 C6×3- 1+2 C32⋊C9 C32×C9 C3×He3 C3×3- 1+2 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 16 2 2 6 16 2 2 6 6 6 6 6 12 12

Matrix representation of C2×C9⋊He3 in GL7(𝔽19)

 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

G:=sub<GL(7,GF(19))| [18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C2×C9⋊He3 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes {\rm He}_3
% in TeX

G:=Group("C2xC9:He3");
// GroupNames label

G:=SmallGroup(486,198);
// by ID

G=gap.SmallGroup(486,198);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,176,2169]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=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,e*b*e^-1=b^7,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽