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G = C2×C324D9order 324 = 22·34

Direct product of C2 and C324D9

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C324D9, C329D18, C33.13D6, C6⋊(C9⋊S3), C18⋊(C3⋊S3), (C3×C6)⋊4D9, (C3×C9)⋊20D6, (C3×C18)⋊11S3, (C32×C18)⋊5C2, (C32×C6).20S3, (C32×C9)⋊10C22, C6.2(C33⋊C2), C32(C2×C9⋊S3), C92(C2×C3⋊S3), C3.(C2×C33⋊C2), (C3×C6).22(C3⋊S3), C32.12(C2×C3⋊S3), SmallGroup(324,149)

Series: Derived Chief Lower central Upper central

C1C32×C9 — C2×C324D9
C1C3C32C33C32×C9C324D9 — C2×C324D9
C32×C9 — C2×C324D9
C1C2

Generators and relations for C2×C324D9
 G = < a,b,c,d,e | a2=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2110 in 250 conjugacy classes, 103 normal (9 characteristic)
C1, C2, C2 [×2], C3, C3 [×12], C22, S3 [×26], C6, C6 [×12], C9 [×9], C32 [×13], D6 [×13], D9 [×18], C18 [×9], C3⋊S3 [×26], C3×C6 [×13], C3×C9 [×12], C33, D18 [×9], C2×C3⋊S3 [×13], C9⋊S3 [×24], C3×C18 [×12], C33⋊C2 [×2], C32×C6, C32×C9, C2×C9⋊S3 [×12], C2×C33⋊C2, C324D9 [×2], C32×C18, C2×C324D9
Quotients: C1, C2 [×3], C22, S3 [×13], D6 [×13], D9 [×9], C3⋊S3 [×13], D18 [×9], C2×C3⋊S3 [×13], C9⋊S3 [×12], C33⋊C2, C2×C9⋊S3 [×12], C2×C33⋊C2, C324D9, C2×C324D9

Smallest permutation representation of C2×C324D9
On 162 points
Generators in S162
(1 134)(2 135)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 89)(29 90)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 91)(44 92)(45 93)(46 126)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 116)(56 117)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(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 22 18)(2 23 10)(3 24 11)(4 25 12)(5 26 13)(6 27 14)(7 19 15)(8 20 16)(9 21 17)(28 158 58)(29 159 59)(30 160 60)(31 161 61)(32 162 62)(33 154 63)(34 155 55)(35 156 56)(36 157 57)(37 52 67)(38 53 68)(39 54 69)(40 46 70)(41 47 71)(42 48 72)(43 49 64)(44 50 65)(45 51 66)(73 115 85)(74 116 86)(75 117 87)(76 109 88)(77 110 89)(78 111 90)(79 112 82)(80 113 83)(81 114 84)(91 120 145)(92 121 146)(93 122 147)(94 123 148)(95 124 149)(96 125 150)(97 126 151)(98 118 152)(99 119 153)(100 141 131)(101 142 132)(102 143 133)(103 144 134)(104 136 135)(105 137 127)(106 138 128)(107 139 129)(108 140 130)
(1 77 41)(2 78 42)(3 79 43)(4 80 44)(5 81 45)(6 73 37)(7 74 38)(8 75 39)(9 76 40)(10 90 72)(11 82 64)(12 83 65)(13 84 66)(14 85 67)(15 86 68)(16 87 69)(17 88 70)(18 89 71)(19 116 53)(20 117 54)(21 109 46)(22 110 47)(23 111 48)(24 112 49)(25 113 50)(26 114 51)(27 115 52)(28 152 144)(29 153 136)(30 145 137)(31 146 138)(32 147 139)(33 148 140)(34 149 141)(35 150 142)(36 151 143)(55 124 100)(56 125 101)(57 126 102)(58 118 103)(59 119 104)(60 120 105)(61 121 106)(62 122 107)(63 123 108)(91 127 160)(92 128 161)(93 129 162)(94 130 154)(95 131 155)(96 132 156)(97 133 157)(98 134 158)(99 135 159)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 135)(9 134)(10 101)(11 100)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 137)(20 136)(21 144)(22 143)(23 142)(24 141)(25 140)(26 139)(27 138)(28 46)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 161)(38 160)(39 159)(40 158)(41 157)(42 156)(43 155)(44 154)(45 162)(55 64)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 92)(74 91)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 126)(90 125)(109 152)(110 151)(111 150)(112 149)(113 148)(114 147)(115 146)(116 145)(117 153)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A···3M6A···6M9A···9AA18A···18AA
order12223···36···69···918···18
size1181812···22···22···22···2

84 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3S3D6D6D9D18
kernelC2×C324D9C324D9C32×C18C3×C18C32×C6C3×C9C33C3×C6C32
# reps1211211212727

Matrix representation of C2×C324D9 in GL6(𝔽19)

100000
010000
0018000
0001800
0000180
0000018
,
010000
18180000
0001800
0011800
000010
000001
,
100000
010000
0018100
0018000
0000181
0000180
,
100000
010000
0071400
005200
0000018
0000118
,
100000
18180000
00141700
0012500
0000181
000001

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,5,0,0,0,0,14,2,0,0,0,0,0,0,0,1,0,0,0,0,18,18],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,14,12,0,0,0,0,17,5,0,0,0,0,0,0,18,0,0,0,0,0,1,1] >;

C2×C324D9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4D_9
% in TeX

G:=Group("C2xC3^2:4D9");
// GroupNames label

G:=SmallGroup(324,149);
// by ID

G=gap.SmallGroup(324,149);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2090,986,579,2164,7781]);
// Polycyclic

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

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×
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