direct product, metacyclic, supersoluble, monomial
Aliases: C3×Dic36, C72.5C6, C24.5D9, C6.19D36, C12.71D18, Dic18.3C6, C32.3Dic12, C8.(C3×D9), (C3×C9)⋊5Q16, C9⋊4(C3×Q16), C4.8(C6×D9), C24.1(C3×S3), (C3×C72).2C2, C2.3(C3×D36), C6.1(C3×D12), C12.64(S3×C6), C36.31(C2×C6), (C3×C24).14S3, (C3×C6).49D12, C18.17(C3×D4), (C3×C18).24D4, (C3×C12).202D6, C3.1(C3×Dic12), (C3×C36).55C22, (C3×Dic18).2C2, SmallGroup(432,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic36
G = < a,b,c | a3=b72=1, c2=b36, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 222 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C24, C24, Dic6, C3×Q8, C3×C9, Dic9, C36, C36, C3×Dic3, C3×C12, Dic12, C3×Q16, C3×C18, C72, C72, Dic18, C3×C24, C3×Dic6, C3×Dic9, C3×C36, Dic36, C3×Dic12, C3×C72, C3×Dic18, C3×Dic36
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, D9, C3×S3, D12, C3×D4, D18, S3×C6, Dic12, C3×Q16, C3×D9, D36, C3×D12, C6×D9, Dic36, C3×Dic12, C3×D36, C3×Dic36
►On 144 points
Generators in S144
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)(73 97 121)(74 98 122)(75 99 123)(76 100 124)(77 101 125)(78 102 126)(79 103 127)(80 104 128)(81 105 129)(82 106 130)(83 107 131)(84 108 132)(85 109 133)(86 110 134)(87 111 135)(88 112 136)(89 113 137)(90 114 138)(91 115 139)(92 116 140)(93 117 141)(94 118 142)(95 119 143)(96 120 144)
(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 95 37 131)(2 94 38 130)(3 93 39 129)(4 92 40 128)(5 91 41 127)(6 90 42 126)(7 89 43 125)(8 88 44 124)(9 87 45 123)(10 86 46 122)(11 85 47 121)(12 84 48 120)(13 83 49 119)(14 82 50 118)(15 81 51 117)(16 80 52 116)(17 79 53 115)(18 78 54 114)(19 77 55 113)(20 76 56 112)(21 75 57 111)(22 74 58 110)(23 73 59 109)(24 144 60 108)(25 143 61 107)(26 142 62 106)(27 141 63 105)(28 140 64 104)(29 139 65 103)(30 138 66 102)(31 137 67 101)(32 136 68 100)(33 135 69 99)(34 134 70 98)(35 133 71 97)(36 132 72 96)
G:=sub<Sym(144)| (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(73,97,121)(74,98,122)(75,99,123)(76,100,124)(77,101,125)(78,102,126)(79,103,127)(80,104,128)(81,105,129)(82,106,130)(83,107,131)(84,108,132)(85,109,133)(86,110,134)(87,111,135)(88,112,136)(89,113,137)(90,114,138)(91,115,139)(92,116,140)(93,117,141)(94,118,142)(95,119,143)(96,120,144), (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), (1,95,37,131)(2,94,38,130)(3,93,39,129)(4,92,40,128)(5,91,41,127)(6,90,42,126)(7,89,43,125)(8,88,44,124)(9,87,45,123)(10,86,46,122)(11,85,47,121)(12,84,48,120)(13,83,49,119)(14,82,50,118)(15,81,51,117)(16,80,52,116)(17,79,53,115)(18,78,54,114)(19,77,55,113)(20,76,56,112)(21,75,57,111)(22,74,58,110)(23,73,59,109)(24,144,60,108)(25,143,61,107)(26,142,62,106)(27,141,63,105)(28,140,64,104)(29,139,65,103)(30,138,66,102)(31,137,67,101)(32,136,68,100)(33,135,69,99)(34,134,70,98)(35,133,71,97)(36,132,72,96)>;
G:=Group( (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(73,97,121)(74,98,122)(75,99,123)(76,100,124)(77,101,125)(78,102,126)(79,103,127)(80,104,128)(81,105,129)(82,106,130)(83,107,131)(84,108,132)(85,109,133)(86,110,134)(87,111,135)(88,112,136)(89,113,137)(90,114,138)(91,115,139)(92,116,140)(93,117,141)(94,118,142)(95,119,143)(96,120,144), (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), (1,95,37,131)(2,94,38,130)(3,93,39,129)(4,92,40,128)(5,91,41,127)(6,90,42,126)(7,89,43,125)(8,88,44,124)(9,87,45,123)(10,86,46,122)(11,85,47,121)(12,84,48,120)(13,83,49,119)(14,82,50,118)(15,81,51,117)(16,80,52,116)(17,79,53,115)(18,78,54,114)(19,77,55,113)(20,76,56,112)(21,75,57,111)(22,74,58,110)(23,73,59,109)(24,144,60,108)(25,143,61,107)(26,142,62,106)(27,141,63,105)(28,140,64,104)(29,139,65,103)(30,138,66,102)(31,137,67,101)(32,136,68,100)(33,135,69,99)(34,134,70,98)(35,133,71,97)(36,132,72,96) );
G=PermutationGroup([[(1,49,25),(2,50,26),(3,51,27),(4,52,28),(5,53,29),(6,54,30),(7,55,31),(8,56,32),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48),(73,97,121),(74,98,122),(75,99,123),(76,100,124),(77,101,125),(78,102,126),(79,103,127),(80,104,128),(81,105,129),(82,106,130),(83,107,131),(84,108,132),(85,109,133),(86,110,134),(87,111,135),(88,112,136),(89,113,137),(90,114,138),(91,115,139),(92,116,140),(93,117,141),(94,118,142),(95,119,143),(96,120,144)], [(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)], [(1,95,37,131),(2,94,38,130),(3,93,39,129),(4,92,40,128),(5,91,41,127),(6,90,42,126),(7,89,43,125),(8,88,44,124),(9,87,45,123),(10,86,46,122),(11,85,47,121),(12,84,48,120),(13,83,49,119),(14,82,50,118),(15,81,51,117),(16,80,52,116),(17,79,53,115),(18,78,54,114),(19,77,55,113),(20,76,56,112),(21,75,57,111),(22,74,58,110),(23,73,59,109),(24,144,60,108),(25,143,61,107),(26,142,62,106),(27,141,63,105),(28,140,64,104),(29,139,65,103),(30,138,66,102),(31,137,67,101),(32,136,68,100),(33,135,69,99),(34,134,70,98),(35,133,71,97),(36,132,72,96)]])
117 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 36 | 36 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | Q16 | D9 | C3×S3 | C3×D4 | D12 | D18 | S3×C6 | C3×Q16 | Dic12 | C3×D9 | D36 | C3×D12 | C6×D9 | Dic36 | C3×Dic12 | C3×D36 | C3×Dic36 |
| kernel | C3×Dic36 | C3×C72 | C3×Dic18 | Dic36 | C72 | Dic18 | C3×C24 | C3×C18 | C3×C12 | C3×C9 | C24 | C24 | C18 | C3×C6 | C12 | C12 | C9 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 2 | 3 | 2 | 4 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C3×Dic36 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 28 | 0 |
| 0 | 60 |
| 0 | 1 |
| 72 | 0 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[28,0,0,60],[0,72,1,0] >;
C3×Dic36 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{36} % in TeX
G:=Group("C3xDic36"); // GroupNames label
G:=SmallGroup(432,104);
// by ID
G=gap.SmallGroup(432,104);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^3=b^72=1,c^2=b^36,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations