C2xC9^2:9C3 - GroupNames

G = C₂×C₉²⋊₉C₃ order 486 = 2·3⁵

Direct product of C₂ and C₉²⋊₉C₃

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

Aliases: C₂×C₉²⋊₉C₃, C₉²⋊₂₂C₆, C₁₈⋊₂3_-¹⁺², C₉⋊C₉⋊₁₂C₆, (C₉×C₁₈)⋊₉C₃, C₃³.₁₄(C₃×C₆), (C₃×C₆).₃₀C₃³, (C₃×C₁₈).₁₃C₃², C₉⋊₅(C₂×3_-¹⁺²), C₃².₃₄(C₃²×C₆), (C₃²×C₆).₁₃C₃², (C₆×3_-¹⁺²).₇C₃, C₆.₁₀(C₃×3_-¹⁺²), C₃.₁₀(C₆×3_-¹⁺²), (C₃×3_-¹⁺²).₁₀C₆, (C₂×C₉⋊C₉)⋊₉C₃, (C₃×C₉).₇(C₃×C₆), SmallGroup(486,206)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₉²⋊₉C₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₉² — C₉²⋊₉C₃ — C₂×C₉²⋊₉C₃

Lower central

C₁ — C₃² — C₂×C₉²⋊₉C₃

Upper central

C₁ — C₃×C₆ — C₂×C₉²⋊₉C₃

Generators and relations for C₂×C₉²⋊₉C₃
G = < a,b,c,d | a²=b⁹=c⁹=d³=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b⁷, dcd^-1=c⁷ >

Subgroups: 252 in 140 conjugacy classes, 90 normal (10 characteristic)
C₁, C₂, C₃, C₃, C₆, C₆, C₉, C₉, C₃², C₃², C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, 3_-¹⁺², C₃³, C₃×C₁₈, C₂×3_-¹⁺², C₃²×C₆, C₉², C₉⋊C₉, C₃×3_-¹⁺², C₉×C₁₈, C₂×C₉⋊C₉, C₆×3_-¹⁺², C₉²⋊₉C₃, C₂×C₉²⋊₉C₃
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, 3_-¹⁺², C₃³, C₂×3_-¹⁺², C₃²×C₆, C₃×3_-¹⁺², C₆×3_-¹⁺², C₉²⋊₉C₃, C₂×C₉²⋊₉C₃

Smallest permutation representation of C₂×C₉²⋊₉C₃
►On 162 points

Generators in S₁₆₂

(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 82)(36 83)(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)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 109)(63 110)(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)

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

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

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

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

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

102 conjugacy classes

class	1	2	3A	···	3H	3I	3J	6A	···	6H	6I	6J	9A	···	9X	9Y	···	9AN	18A	···	18X	18Y	···	18AN
order	1	2	3	···	3	3	3	6	···	6	6	6	9	···	9	9	···	9	18	···	18	18	···	18
size	1	1	1	···	1	9	9	1	···	1	9	9	3	···	3	9	···	9	3	···	3	9	···	9

102 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	3_-¹⁺²	C₂×3_-¹⁺²
kernel	C₂×C₉²⋊₉C₃	C₉²⋊₉C₃	C₉×C₁₈	C₂×C₉⋊C₉	C₆×3_-¹⁺²	C₉²	C₉⋊C₉	C₃×3_-¹⁺²	C₁₈	C₉
# reps	1	1	2	16	8	2	16	8	24	24

Matrix representation of C₂×C₉²⋊₉C₃ ►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

1	0	0	0	0	0	0
0	0	11	0	0	0	0
0	18	12	9	0	0	0
0	11	0	7	0	0	0
0	0	0	0	16	10	0
0	0	0	0	5	3	1
0	0	0	0	15	17	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	12	8	6	0	0	0
0	1	0	11	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	1	0	0	0	0	0
0	0	11	0	0	0	0
0	12	1	7	0	0	0
0	0	0	0	1	0	0
0	0	0	0	2	7	0
0	0	0	0	2	0	11

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],[1,0,0,0,0,0,0,0,0,18,11,0,0,0,0,11,12,0,0,0,0,0,0,9,7,0,0,0,0,0,0,0,16,5,15,0,0,0,0,10,3,17,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,1,8,0,0,0,0,0,0,6,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,7,0,0,0,0,0,0,0,11] >;

C₂×C₉²⋊₉C₃ in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_9C_3

% in TeX

G:=Group("C2xC9^2:9C3");

// GroupNames label

G:=SmallGroup(486,206);

// by ID

G=gap.SmallGroup(486,206);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,176,2169,237]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^7,d*c*d^-1=c^7>;

// generators/relations

G = C2×C92⋊9C3 order 486 = 2·35

Direct product of C2 and C92⋊9C3

G = C₂×C₉²⋊₉C₃ order 486 = 2·3⁵

Direct product of C₂ and C₉²⋊₉C₃