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## G = C9×D6⋊C4order 432 = 24·33

### Direct product of C9 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×D6⋊C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C18 — S3×C2×C18 — C9×D6⋊C4
 Lower central C3 — C6 — C9×D6⋊C4
 Upper central C1 — C2×C18 — C2×C36

Generators and relations for C9×D6⋊C4
G = < a,b,c,d | a9=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 260 in 118 conjugacy classes, 51 normal (45 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, D6⋊C4, C3×C22⋊C4, S3×C9, C3×C18, C2×C36, C2×C36, C22×C18, C6×Dic3, C6×C12, S3×C2×C6, C9×Dic3, C3×C36, S3×C18, S3×C18, C6×C18, C9×C22⋊C4, C3×D6⋊C4, Dic3×C18, C6×C36, S3×C2×C18, C9×D6⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C9, C12, D6, C2×C6, C22⋊C4, C18, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C36, C2×C18, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C9, C2×C36, D4×C9, S3×C12, C3×D12, C3×C3⋊D4, S3×C18, C9×C22⋊C4, C3×D6⋊C4, S3×C36, C9×D12, C9×C3⋊D4, C9×D6⋊C4

Smallest permutation representation of C9×D6⋊C4
On 144 points
Generators in S144
(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 43 4 37 7 40)(2 44 5 38 8 41)(3 45 6 39 9 42)(10 143 16 140 13 137)(11 144 17 141 14 138)(12 136 18 142 15 139)(19 135 25 132 22 129)(20 127 26 133 23 130)(21 128 27 134 24 131)(28 51 31 54 34 48)(29 52 32 46 35 49)(30 53 33 47 36 50)(55 86 58 89 61 83)(56 87 59 90 62 84)(57 88 60 82 63 85)(64 73 67 76 70 79)(65 74 68 77 71 80)(66 75 69 78 72 81)(91 119 97 125 94 122)(92 120 98 126 95 123)(93 121 99 118 96 124)(100 115 106 112 103 109)(101 116 107 113 104 110)(102 117 108 114 105 111)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 71)(11 72)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 63)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 123)(29 124)(30 125)(31 126)(32 118)(33 119)(34 120)(35 121)(36 122)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 109)(44 110)(45 111)(46 99)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 136)(80 137)(81 138)(82 135)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)
(1 63 35 79)(2 55 36 80)(3 56 28 81)(4 57 29 73)(5 58 30 74)(6 59 31 75)(7 60 32 76)(8 61 33 77)(9 62 34 78)(10 104 127 125)(11 105 128 126)(12 106 129 118)(13 107 130 119)(14 108 131 120)(15 100 132 121)(16 101 133 122)(17 102 134 123)(18 103 135 124)(19 96 136 112)(20 97 137 113)(21 98 138 114)(22 99 139 115)(23 91 140 116)(24 92 141 117)(25 93 142 109)(26 94 143 110)(27 95 144 111)(37 88 52 67)(38 89 53 68)(39 90 54 69)(40 82 46 70)(41 83 47 71)(42 84 48 72)(43 85 49 64)(44 86 50 65)(45 87 51 66)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 9A ··· 9F 9G ··· 9L 12A ··· 12P 12Q 12R 12S 12T 18A ··· 18R 18S ··· 18AJ 18AK ··· 18AV 36A ··· 36AJ 36AK ··· 36AV order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 9 ··· 9 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 6 6 1 1 2 2 2 2 2 6 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C9 C12 C18 C18 C18 C36 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C9 D4×C9 S3×C12 C3×D12 C3×C3⋊D4 S3×C18 S3×C36 C9×D12 C9×C3⋊D4 kernel C9×D6⋊C4 Dic3×C18 C6×C36 S3×C2×C18 C3×D6⋊C4 S3×C18 C6×Dic3 C6×C12 S3×C2×C6 D6⋊C4 S3×C6 C2×Dic3 C2×C12 C22×S3 D6 C2×C36 C3×C18 C2×C18 C2×C12 C18 C18 C18 C3×C6 C2×C6 C2×C4 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 6 8 6 6 6 24 1 2 1 2 2 2 2 4 2 6 12 4 4 4 6 12 12 12

Matrix representation of C9×D6⋊C4 in GL4(𝔽37) generated by

 7 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 26 0 0 0 11 10 0 0 0 0 11 25 0 0 0 27
,
 27 28 0 0 11 10 0 0 0 0 9 19 0 0 25 28
,
 6 0 0 0 0 6 0 0 0 0 6 9 0 0 0 31
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[26,11,0,0,0,10,0,0,0,0,11,0,0,0,25,27],[27,11,0,0,28,10,0,0,0,0,9,25,0,0,19,28],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,9,31] >;

C9×D6⋊C4 in GAP, Magma, Sage, TeX

C_9\times D_6\rtimes C_4
% in TeX

G:=Group("C9xD6:C4");
// GroupNames label

G:=SmallGroup(432,135);
// by ID

G=gap.SmallGroup(432,135);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,268,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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