C2xC9^2:5C3 - GroupNames

G = C₂×C₉²⋊₅C₃ order 486 = 2·3⁵

Direct product of C₂ and C₉²⋊₅C₃

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

Aliases: C₂×C₉²⋊₅C₃, C₉²⋊₁₈C₆, C₉⋊C₉⋊₁₀C₆, (C₉×C₁₈)⋊₅C₃, C₃²⋊C₉.₂₁C₆, (C₃×C₆).₂₈C₃³, C₃³.₁₂(C₃×C₆), C₆.₁₀(C₉○He₃), (C₃×C₁₈).₁₁C₃², C₃².₃₂(C₃²×C₆), (C₃²×C₆).₁₁C₃², (C₂×C₉⋊C₉)⋊₇C₃, (C₃×C₉).₂₈(C₃×C₆), C₃.₁₀(C₂×C₉○He₃), (C₂×C₃²⋊C₉).₁₂C₃, SmallGroup(486,204)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₉²⋊₅C₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₉² — C₉²⋊₅C₃ — C₂×C₉²⋊₅C₃

Lower central

C₁ — C₃² — C₂×C₉²⋊₅C₃

Upper central

C₁ — C₃×C₆ — C₂×C₉²⋊₅C₃

Generators and relations for C₂×C₉²⋊₅C₃
G = < a,b,c,d | a²=b⁹=c⁹=d³=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=bc³, dcd^-1=b⁶c >

Subgroups: 180 in 100 conjugacy classes, 66 normal (10 characteristic)
C₁, C₂, C₃, C₃, C₆, C₆, C₉, C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, C₃³, C₃×C₁₈, C₃²×C₆, C₉², C₃²⋊C₉, C₉⋊C₉, C₉×C₁₈, C₂×C₃²⋊C₉, C₂×C₉⋊C₉, C₉²⋊₅C₃, C₂×C₉²⋊₅C₃
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, C₃³, C₃²×C₆, C₉○He₃, C₂×C₉○He₃, C₉²⋊₅C₃, C₂×C₉²⋊₅C₃

Smallest permutation representation of C₂×C₉²⋊₅C₃
►On 162 points

Generators in S₁₆₂

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

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

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

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

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

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	C₉○He₃	C₂×C₉○He₃
kernel	C₂×C₉²⋊₅C₃	C₉²⋊₅C₃	C₉×C₁₈	C₂×C₃²⋊C₉	C₂×C₉⋊C₉	C₉²	C₃²⋊C₉	C₉⋊C₉	C₆	C₃
# reps	1	1	2	8	16	2	8	16	24	24

Matrix representation of C₂×C₉²⋊₅C₃ ►in GL₆(𝔽₁₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	0	0	18

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	17	1	0
0	0	0	0	2	16
0	0	0	14	14	0

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	7	6	0
0	0	0	0	12	1
0	0	0	8	8	0

1	0	0	0	0	0
0	7	0	0	0	0
0	0	11	0	0	0
0	0	0	1	0	0
0	0	0	18	11	0
0	0	0	11	0	7

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,17,0,14,0,0,0,1,2,14,0,0,0,0,16,0],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,8,0,0,0,6,12,8,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,18,11,0,0,0,0,11,0,0,0,0,0,0,7] >;

C₂×C₉²⋊₅C₃ in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_5C_3

% in TeX

G:=Group("C2xC9^2:5C3");

// GroupNames label

G:=SmallGroup(486,204);

// by ID

G=gap.SmallGroup(486,204);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,500,2169,165]);

// 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*c^3,d*c*d^-1=b^6*c>;

// generators/relations

G = C2×C92⋊5C3 order 486 = 2·35

Direct product of C2 and C92⋊5C3

G = C₂×C₉²⋊₅C₃ order 486 = 2·3⁵

Direct product of C₂ and C₉²⋊₅C₃