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## G = C2×C6×Dic9order 432 = 24·33

### Direct product of C2×C6 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C6×Dic9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C3×Dic9 — C6×Dic9 — C2×C6×Dic9
 Lower central C9 — C2×C6×Dic9
 Upper central C1 — C22×C6

Generators and relations for C2×C6×Dic9
G = < a,b,c,d | a2=b6=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 446 in 194 conjugacy classes, 118 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×C18, C2×C6×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, D9, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, Dic9, D18, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C3×D9, C2×Dic9, C22×D9, C6×Dic3, S3×C2×C6, C3×Dic9, C6×D9, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×D9, C2×C6×Dic9

Smallest permutation representation of C2×C6×Dic9
On 144 points
Generators in S144
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 125)(74 126)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 127)(108 128)
(1 34 13 28 7 22)(2 35 14 29 8 23)(3 36 15 30 9 24)(4 19 16 31 10 25)(5 20 17 32 11 26)(6 21 18 33 12 27)(37 70 49 64 43 58)(38 71 50 65 44 59)(39 72 51 66 45 60)(40 55 52 67 46 61)(41 56 53 68 47 62)(42 57 54 69 48 63)(73 103 79 91 85 97)(74 104 80 92 86 98)(75 105 81 93 87 99)(76 106 82 94 88 100)(77 107 83 95 89 101)(78 108 84 96 90 102)(109 143 115 131 121 137)(110 144 116 132 122 138)(111 127 117 133 123 139)(112 128 118 134 124 140)(113 129 119 135 125 141)(114 130 120 136 126 142)
(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 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 97 28 106)(20 96 29 105)(21 95 30 104)(22 94 31 103)(23 93 32 102)(24 92 33 101)(25 91 34 100)(26 108 35 99)(27 107 36 98)(37 113 46 122)(38 112 47 121)(39 111 48 120)(40 110 49 119)(41 109 50 118)(42 126 51 117)(43 125 52 116)(44 124 53 115)(45 123 54 114)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A ··· 4H 6A ··· 6N 6O ··· 6AI 9A ··· 9I 12A ··· 12P 18A ··· 18BK order 1 2 ··· 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 ··· 1 1 1 2 2 2 9 ··· 9 1 ··· 1 2 ··· 2 2 ··· 2 9 ··· 9 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 D9 C3×S3 Dic9 D18 C3×Dic3 S3×C6 C3×D9 C3×Dic9 C6×D9 kernel C2×C6×Dic9 C6×Dic9 C2×C6×C18 C22×Dic9 C6×C18 C2×Dic9 C22×C18 C2×C18 C2×C62 C62 C62 C22×C6 C22×C6 C2×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 8 12 2 16 1 4 3 3 2 12 9 8 6 6 24 18

Matrix representation of C2×C6×Dic9 in GL4(𝔽37) generated by

 1 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 11 0 0 0 0 27 0 0 0 0 10 0 0 0 0 10
,
 36 0 0 0 0 1 0 0 0 0 16 0 0 0 14 7
,
 31 0 0 0 0 1 0 0 0 0 23 9 0 0 3 14
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[11,0,0,0,0,27,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,16,14,0,0,0,7],[31,0,0,0,0,1,0,0,0,0,23,3,0,0,9,14] >;

C2×C6×Dic9 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_9
% in TeX

G:=Group("C2xC6xDic9");
// GroupNames label

G:=SmallGroup(432,372);
// by ID

G=gap.SmallGroup(432,372);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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