direct product, metacyclic, supersoluble, monomial
Aliases: C3×C4⋊Dic9, C36⋊5C12, C12⋊3Dic9, C6.22D36, C6.10Dic18, C62.120D6, C4⋊(C3×Dic9), (C3×C36)⋊2C4, (C6×C36).7C2, C2.1(C3×D36), (C3×C18).7Q8, C18.5(C3×Q8), (C2×C36).16C6, (C6×C12).41S3, (C2×C12).19D9, (C2×C6).48D18, C6.10(C3×D12), (C3×C18).27D4, C18.20(C3×D4), (C3×C6).52D12, C6.9(C6×Dic3), C6.4(C3×Dic6), C2.4(C6×Dic9), C22.5(C6×D9), C18.22(C2×C12), (C2×Dic9).6C6, (C6×Dic9).6C2, C6.20(C2×Dic9), (C3×C6).23Dic6, C12.2(C3×Dic3), C2.2(C3×Dic18), (C6×C18).34C22, (C3×C12).19Dic3, C32.3(C4⋊Dic3), C9⋊5(C3×C4⋊C4), (C3×C9)⋊7(C4⋊C4), (C2×C4).3(C3×D9), (C2×C6).36(S3×C6), C3.1(C3×C4⋊Dic3), (C2×C12).11(C3×S3), (C2×C18).25(C2×C6), (C3×C18).34(C2×C4), (C3×C6).57(C2×Dic3), SmallGroup(432,130)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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