direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic9⋊C4, Dic9⋊3C12, C6.9Dic18, C62.119D6, (C6×C36).2C2, (C2×C12).3D9, C6.24(C4×D9), C2.4(C12×D9), C18.4(C3×Q8), (C3×C18).6Q8, C6.12(S3×C12), (C2×C36).14C6, (C6×C12).23S3, (C3×Dic9)⋊3C4, (C3×C18).39D4, (C2×C6).47D18, C18.22(C3×D4), C6.3(C3×Dic6), C22.4(C6×D9), C18.18(C2×C12), C6.29(C9⋊D4), (C2×Dic9).5C6, (C6×Dic9).5C2, (C3×C6).22Dic6, C2.1(C3×Dic18), (C6×C18).33C22, C32.4(Dic3⋊C4), C9⋊4(C3×C4⋊C4), (C3×C9)⋊6(C4⋊C4), (C2×C4).1(C3×D9), C2.1(C3×C9⋊D4), (C2×C6).35(S3×C6), (C3×C6).66(C4×S3), (C2×C12).1(C3×S3), C6.12(C3×C3⋊D4), (C3×C18).21(C2×C4), (C2×C18).24(C2×C6), C3.1(C3×Dic3⋊C4), (C3×C6).90(C3⋊D4), SmallGroup(432,129)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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