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G = C2×D6⋊D9order 432 = 24·33

Direct product of C2 and D6⋊D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C2×D6⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — D6⋊D9 — C2×D6⋊D9
 Lower central C3×C9 — C3×C18 — C2×D6⋊D9
 Upper central C1 — C22

Generators and relations for C2×D6⋊D9
G = < a,b,c,d,e | a2=b6=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 956 in 178 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, D6, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×S3, C22×C6, C3×C9, Dic9, D18, D18, C2×C18, C2×C18, C3⋊Dic3, S3×C6, S3×C6, C62, C2×C3⋊D4, C3×D9, S3×C9, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C22×D9, C22×C18, D6⋊S3, C2×C3⋊Dic3, S3×C2×C6, S3×C2×C6, C9⋊Dic3, C6×D9, C6×D9, S3×C18, S3×C18, C6×C18, C2×C9⋊D4, C2×D6⋊S3, D6⋊D9, C2×C9⋊Dic3, C2×C6×D9, S3×C2×C18, C2×D6⋊D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, S32, C2×C3⋊D4, C9⋊D4, C22×D9, D6⋊S3, C2×S32, S3×D9, C2×C9⋊D4, C2×D6⋊S3, D6⋊D9, C2×S3×D9, C2×D6⋊D9

Smallest permutation representation of C2×D6⋊D9
On 144 points
Generators in S144
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 127)(119 128)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)
(1 16 4 10 7 13)(2 17 5 11 8 14)(3 18 6 12 9 15)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)(37 49 43 46 40 52)(38 50 44 47 41 53)(39 51 45 48 42 54)(55 67 61 64 58 70)(56 68 62 65 59 71)(57 69 63 66 60 72)(73 85 79 82 76 88)(74 86 80 83 77 89)(75 87 81 84 78 90)(91 103 97 100 94 106)(92 104 98 101 95 107)(93 105 99 102 96 108)(109 124 112 118 115 121)(110 125 113 119 116 122)(111 126 114 120 117 123)(127 142 130 136 133 139)(128 143 131 137 134 140)(129 144 132 138 135 141)
(1 88)(2 89)(3 90)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 79)(11 80)(12 81)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 106)(20 107)(21 108)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 97)(29 98)(30 99)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 118)(44 119)(45 120)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 109)(53 110)(54 111)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 136)(62 137)(63 138)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 127)(71 128)(72 129)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 63)(9 62)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 72)(18 71)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 45)(27 44)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 54)(36 53)(73 142)(74 141)(75 140)(76 139)(77 138)(78 137)(79 136)(80 144)(81 143)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 135)(90 134)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 126)(99 125)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 117)(108 116)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 9A 9B 9C 9D 9E 9F 18A ··· 18I 18J ··· 18R 18S ··· 18AD order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 ··· 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 1 1 6 6 18 18 2 2 4 54 54 2 ··· 2 4 4 4 6 6 6 6 18 18 18 18 2 2 2 4 4 4 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 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 S3 S3 D4 D6 D6 D6 D6 D9 C3⋊D4 C3⋊D4 D18 D18 C9⋊D4 S32 D6⋊S3 C2×S32 S3×D9 D6⋊D9 C2×S3×D9 kernel C2×D6⋊D9 D6⋊D9 C2×C9⋊Dic3 C2×C6×D9 S3×C2×C18 C22×D9 S3×C2×C6 C3×C18 D18 C2×C18 S3×C6 C62 C22×S3 C18 C3×C6 D6 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 1 2 2 1 2 1 3 4 4 6 3 12 1 2 1 3 6 3

Matrix representation of C2×D6⋊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 36 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 30 14 0 0 0 0 23 7 0 0 0 0 0 0 36 36 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 0 36 0 0 0 0 1 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 27 0 0 0 0 3 11
,
 26 17 0 0 0 0 6 11 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 31 27 0 0 0 0 22 6

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,23,0,0,0,0,14,7,0,0,0,0,0,0,36,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,3,0,0,0,0,27,11],[26,6,0,0,0,0,17,11,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,31,22,0,0,0,0,27,6] >;

C2×D6⋊D9 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes D_9
% in TeX

G:=Group("C2xD6:D9");
// GroupNames label

G:=SmallGroup(432,311);
// by ID

G=gap.SmallGroup(432,311);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,3091,662,4037,7069]);
// Polycyclic

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

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