C2xC9:He3 - GroupNames

G = C₂×C₉⋊He₃ order 486 = 2·3⁵

Direct product of C₂ and C₉⋊He₃

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

Aliases: C₂×C₉⋊He₃, C₁₈⋊He₃, C₉⋊₂(C₂×He₃), C₃.₅(C₆×He₃), C₆.₅(C₃×He₃), C₃²⋊C₉⋊₁₅C₆, (C₃²×C₉)⋊₃₃C₆, (C₃²×C₁₈)⋊₇C₃, (C₆×He₃).₄C₃, C₃³.₈(C₃×C₆), (C₃×He₃).₁₇C₆, C₆.₄(C₉○He₃), (C₃×C₆).₂₂C₃³, (C₃×C₁₈).₅C₃², (C₃×C₆)⋊₁3_-¹⁺², C₃².₂₆(C₃²×C₆), (C₃²×C₆).₂₈C₃², (C₆×3_-¹⁺²)⋊₃C₃, C₆.₅(C₃×3_-¹⁺²), C₃.₅(C₆×3_-¹⁺²), (C₃×3_-¹⁺²)⋊₁₀C₆, C₃²⋊₃(C₂×3_-¹⁺²), (C₃×C₉).₅(C₃×C₆), (C₂×C₃²⋊C₉)⋊₇C₃, C₃.₄(C₂×C₉○He₃), SmallGroup(486,198)

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

G = C2×C9⋊He3 order 486 = 2·35

Direct product of C2 and C9⋊He3

G = C₂×C₉⋊He₃ order 486 = 2·3⁵

Direct product of C₂ and C₉⋊He₃