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## G = C3×Dic9⋊C4order 432 = 24·33

### Direct product of C3 and Dic9⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Dic9⋊C4
G = < a,b,c,d | a3=b18=d4=1, c2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b9c >

Subgroups: 262 in 94 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C3×C18, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C3×Dic9, C3×Dic9, C3×C36, C6×C18, Dic9⋊C4, C3×Dic3⋊C4, C6×Dic9, C6×C36, C3×Dic9⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, D9, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, D18, S3×C6, Dic3⋊C4, C3×C4⋊C4, C3×D9, Dic18, C4×D9, C9⋊D4, C3×Dic6, S3×C12, C3×C3⋊D4, C6×D9, Dic9⋊C4, C3×Dic3⋊C4, C3×Dic18, C12×D9, C3×C9⋊D4, C3×Dic9⋊C4

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

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

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

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

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 D6 D9 C3×S3 C3×D4 C3×Q8 Dic6 C4×S3 C3⋊D4 D18 S3×C6 C3×D9 Dic18 C4×D9 C9⋊D4 C3×Dic6 S3×C12 C3×C3⋊D4 C6×D9 C3×Dic18 C12×D9 C3×C9⋊D4 kernel C3×Dic9⋊C4 C6×Dic9 C6×C36 Dic9⋊C4 C3×Dic9 C2×Dic9 C2×C36 Dic9 C6×C12 C3×C18 C3×C18 C62 C2×C12 C2×C12 C18 C18 C3×C6 C3×C6 C3×C6 C2×C6 C2×C6 C2×C4 C6 C6 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 1 3 2 2 2 2 2 2 3 2 6 6 6 6 4 4 4 6 12 12 12

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

 26 0 0 0 10 0 0 0 10
,
 1 0 0 0 28 0 0 0 4
,
 1 0 0 0 0 1 0 36 0
,
 6 0 0 0 36 0 0 0 1
G:=sub<GL(3,GF(37))| [26,0,0,0,10,0,0,0,10],[1,0,0,0,28,0,0,0,4],[1,0,0,0,0,36,0,1,0],[6,0,0,0,36,0,0,0,1] >;

C3×Dic9⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic9:C4");
// GroupNames label

G:=SmallGroup(432,129);
// by ID

G=gap.SmallGroup(432,129);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,10085,292,14118]);
// Polycyclic

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

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