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## G = C3×C4⋊Dic9order 432 = 24·33

### Direct product of C3 and C4⋊Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×C4⋊Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C6×C18 — C6×Dic9 — C3×C4⋊Dic9
 Lower central C9 — C18 — C3×C4⋊Dic9
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C4⋊Dic9
G = < a,b,c,d | a3=b4=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 262 in 94 conjugacy classes, 54 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C3×C18, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C3×Dic9, C3×C36, C6×C18, C4⋊Dic9, C3×C4⋊Dic3, C6×Dic9, C6×C36, C3×C4⋊Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, D9, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, Dic9, D18, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×D9, Dic18, D36, C2×Dic9, C3×Dic6, C3×D12, C6×Dic3, C3×Dic9, C6×D9, C4⋊Dic9, C3×C4⋊Dic3, C3×Dic18, C3×D36, C6×Dic9, C3×C4⋊Dic9

Smallest permutation representation of C3×C4⋊Dic9
On 144 points
Generators in S144
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)(109 121 115)(110 122 116)(111 123 117)(112 124 118)(113 125 119)(114 126 120)(127 133 139)(128 134 140)(129 135 141)(130 136 142)(131 137 143)(132 138 144)
(1 93 138 72)(2 94 139 55)(3 95 140 56)(4 96 141 57)(5 97 142 58)(6 98 143 59)(7 99 144 60)(8 100 127 61)(9 101 128 62)(10 102 129 63)(11 103 130 64)(12 104 131 65)(13 105 132 66)(14 106 133 67)(15 107 134 68)(16 108 135 69)(17 91 136 70)(18 92 137 71)(19 74 121 44)(20 75 122 45)(21 76 123 46)(22 77 124 47)(23 78 125 48)(24 79 126 49)(25 80 109 50)(26 81 110 51)(27 82 111 52)(28 83 112 53)(29 84 113 54)(30 85 114 37)(31 86 115 38)(32 87 116 39)(33 88 117 40)(34 89 118 41)(35 90 119 42)(36 73 120 43)
(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 38 10 47)(2 37 11 46)(3 54 12 45)(4 53 13 44)(5 52 14 43)(6 51 15 42)(7 50 16 41)(8 49 17 40)(9 48 18 39)(19 57 28 66)(20 56 29 65)(21 55 30 64)(22 72 31 63)(23 71 32 62)(24 70 33 61)(25 69 34 60)(26 68 35 59)(27 67 36 58)(73 142 82 133)(74 141 83 132)(75 140 84 131)(76 139 85 130)(77 138 86 129)(78 137 87 128)(79 136 88 127)(80 135 89 144)(81 134 90 143)(91 117 100 126)(92 116 101 125)(93 115 102 124)(94 114 103 123)(95 113 104 122)(96 112 105 121)(97 111 106 120)(98 110 107 119)(99 109 108 118)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 9A ··· 9I 12A ··· 12P 12Q ··· 12X 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 2 2 2 2 2 18 18 18 18 1 ··· 1 2 ··· 2 2 ··· 2 2 ··· 2 18 ··· 18 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 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 2 2 2 2 2 2 type + + + + + - - + + - + - + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 Dic3 D6 D9 C3×S3 C3×D4 C3×Q8 Dic6 D12 Dic9 C3×Dic3 D18 S3×C6 C3×D9 Dic18 D36 C3×Dic6 C3×D12 C3×Dic9 C6×D9 C3×Dic18 C3×D36 kernel C3×C4⋊Dic9 C6×Dic9 C6×C36 C4⋊Dic9 C3×C36 C2×Dic9 C2×C36 C36 C6×C12 C3×C18 C3×C18 C3×C12 C62 C2×C12 C2×C12 C18 C18 C3×C6 C3×C6 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 2 1 3 2 2 2 2 2 6 4 3 2 6 6 6 4 4 12 6 12 12

Matrix representation of C3×C4⋊Dic9 in GL3(𝔽37) generated by

 10 0 0 0 26 0 0 0 26
,
 1 0 0 0 6 0 0 0 31
,
 36 0 0 0 30 0 0 0 21
,
 31 0 0 0 0 1 0 36 0
G:=sub<GL(3,GF(37))| [10,0,0,0,26,0,0,0,26],[1,0,0,0,6,0,0,0,31],[36,0,0,0,30,0,0,0,21],[31,0,0,0,0,36,0,1,0] >;

C3×C4⋊Dic9 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes {\rm Dic}_9
% in TeX

G:=Group("C3xC4:Dic9");
// GroupNames label

G:=SmallGroup(432,130);
// by ID

G=gap.SmallGroup(432,130);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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