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## G = C2×Dic3×D9order 432 = 24·33

### Direct product of C2, Dic3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×Dic3×D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — Dic3×D9 — C2×Dic3×D9
 Lower central C3×C9 — C2×Dic3×D9
 Upper central C1 — C22

Generators and relations for C2×Dic3×D9
G = < a,b,c,d,e | a2=b6=d9=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 860 in 178 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S3×C6, C62, S3×C2×C4, C22×Dic3, C3×D9, C3×C18, C3×C18, C4×D9, C2×Dic9, C2×C36, C22×D9, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C9×Dic3, C9⋊Dic3, C6×D9, C6×C18, C2×C4×D9, C2×S3×Dic3, Dic3×D9, Dic3×C18, C2×C9⋊Dic3, C2×C6×D9, C2×Dic3×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C4×S3, C2×Dic3, C22×S3, D18, S32, S3×C2×C4, C22×Dic3, C4×D9, C22×D9, S3×Dic3, C2×S32, S3×D9, C2×C4×D9, C2×S3×Dic3, Dic3×D9, C2×S3×D9, C2×Dic3×D9

Smallest permutation representation of C2×Dic3×D9
On 144 points
Generators in S144
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 17 7 14 4 11)(2 18 8 15 5 12)(3 10 9 16 6 13)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 36)(37 52 40 46 43 49)(38 53 41 47 44 50)(39 54 42 48 45 51)(55 70 58 64 61 67)(56 71 59 65 62 68)(57 72 60 66 63 69)(73 88 76 82 79 85)(74 89 77 83 80 86)(75 90 78 84 81 87)(91 106 94 100 97 103)(92 107 95 101 98 104)(93 108 96 102 99 105)(109 121 115 118 112 124)(110 122 116 119 113 125)(111 123 117 120 114 126)(127 139 133 136 130 142)(128 140 134 137 131 143)(129 141 135 138 132 144)
(1 86 14 77)(2 87 15 78)(3 88 16 79)(4 89 17 80)(5 90 18 81)(6 82 10 73)(7 83 11 74)(8 84 12 75)(9 85 13 76)(19 100 28 91)(20 101 29 92)(21 102 30 93)(22 103 31 94)(23 104 32 95)(24 105 33 96)(25 106 34 97)(26 107 35 98)(27 108 36 99)(37 118 46 109)(38 119 47 110)(39 120 48 111)(40 121 49 112)(41 122 50 113)(42 123 51 114)(43 124 52 115)(44 125 53 116)(45 126 54 117)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 139 67 130)(59 140 68 131)(60 141 69 132)(61 142 70 133)(62 143 71 134)(63 144 72 135)
(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)
(1 48)(2 47)(3 46)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 72)(27 71)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 63)(36 62)(73 124)(74 123)(75 122)(76 121)(77 120)(78 119)(79 118)(80 126)(81 125)(82 115)(83 114)(84 113)(85 112)(86 111)(87 110)(88 109)(89 117)(90 116)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 144)(99 143)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 135)(108 134)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 9 9 9 9 2 2 4 3 3 3 3 27 27 27 27 2 ··· 2 4 4 4 18 18 18 18 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C4 S3 S3 Dic3 D6 D6 D6 D6 D9 C4×S3 D18 D18 C4×D9 S32 S3×Dic3 C2×S32 S3×D9 Dic3×D9 C2×S3×D9 kernel C2×Dic3×D9 Dic3×D9 Dic3×C18 C2×C9⋊Dic3 C2×C6×D9 C6×D9 C22×D9 C6×Dic3 D18 D18 C2×C18 C3×Dic3 C62 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 1 4 2 1 2 1 3 4 6 3 12 1 2 1 3 6 3

Matrix representation of C2×Dic3×D9 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 36 1 0 0 0 0 0 0 0 36 0 0 0 0 1 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 31 0 0 0 0 31 0 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 20 0 0 0 0 17 31
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 11 0 0 0 0 17 31

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,31,0,0,0,0,31,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,17,0,0,0,0,20,31],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,17,0,0,0,0,11,31] >;

C2×Dic3×D9 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\times D_9
% in TeX

G:=Group("C2xDic3xD9");
// GroupNames label

G:=SmallGroup(432,304);
// by ID

G=gap.SmallGroup(432,304);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^9=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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