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G = C2×C32⋊4D9  order 324 = 22·34

Direct product of C2 and C32⋊4D9

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

Aliases: C2×C32⋊4D9, C32⋊9D18, 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), C3⋊2(C2×C9⋊S3), C9⋊2(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

Derived series
C1 — C32×C9 — C2×C32⋊4D9
Chief series
C1 — C3 — C32 — C33 — C32×C9 — C32⋊4D9 — C2×C32⋊4D9
Lower central
C32×C9 — C2×C32⋊4D9
Upper central
C1 — C2

Generators and relations for C2×C32⋊4D9
 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, C3, C3, C22, S3, C6, C6, C9, C32, D6, D9, C18, C3⋊S3, C3×C6, C3×C9, C33, D18, C2×C3⋊S3, C9⋊S3, C3×C18, C33⋊C2, C32×C6, C32×C9, C2×C9⋊S3, C2×C33⋊C2, C32⋊4D9, C32×C18, C2×C32⋊4D9
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, C2×C3⋊S3, C9⋊S3, C33⋊C2, C2×C9⋊S3, C2×C33⋊C2, C32⋊4D9, C2×C32⋊4D9

Smallest permutation representation of C2×C32⋊4D9
►On 162 points
Generators in S162
(1 131)(2 132)(3 133)(4 134)(5 135)(6 127)(7 128)(8 129)(9 130)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 149)(20 150)(21 151)(22 152)(23 153)(24 145)(25 146)(26 147)(27 148)(28 89)(29 90)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 104)(47 105)(48 106)(49 107)(50 108)(51 100)(52 101)(53 102)(54 103)(55 116)(56 117)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 141)(65 142)(66 143)(67 144)(68 136)(69 137)(70 138)(71 139)(72 140)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)

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

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

(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)

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

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×C32⋊4D9C32⋊4D9C32×C18C3×C18C32×C6C3×C9C33C3×C6C32
# reps1211211212727

Matrix representation of C2×C32⋊4D9 ►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×C32⋊4D9 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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