metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic36, C8.D9, C9⋊1Q16, C72.1C2, C24.1S3, C4.8D18, C18.1D4, C6.1D12, C2.3D36, C3.Dic12, C12.40D6, C36.8C22, Dic18.1C2, SmallGroup(144,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic36
G = < a,b | a72=1, b2=a36, bab-1=a-1 >
Smallest permutation representation of Dic36
►Regular action on 144 points
Generators in S144
(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 73 37 109)(2 144 38 108)(3 143 39 107)(4 142 40 106)(5 141 41 105)(6 140 42 104)(7 139 43 103)(8 138 44 102)(9 137 45 101)(10 136 46 100)(11 135 47 99)(12 134 48 98)(13 133 49 97)(14 132 50 96)(15 131 51 95)(16 130 52 94)(17 129 53 93)(18 128 54 92)(19 127 55 91)(20 126 56 90)(21 125 57 89)(22 124 58 88)(23 123 59 87)(24 122 60 86)(25 121 61 85)(26 120 62 84)(27 119 63 83)(28 118 64 82)(29 117 65 81)(30 116 66 80)(31 115 67 79)(32 114 68 78)(33 113 69 77)(34 112 70 76)(35 111 71 75)(36 110 72 74)
G:=sub<Sym(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), (1,73,37,109)(2,144,38,108)(3,143,39,107)(4,142,40,106)(5,141,41,105)(6,140,42,104)(7,139,43,103)(8,138,44,102)(9,137,45,101)(10,136,46,100)(11,135,47,99)(12,134,48,98)(13,133,49,97)(14,132,50,96)(15,131,51,95)(16,130,52,94)(17,129,53,93)(18,128,54,92)(19,127,55,91)(20,126,56,90)(21,125,57,89)(22,124,58,88)(23,123,59,87)(24,122,60,86)(25,121,61,85)(26,120,62,84)(27,119,63,83)(28,118,64,82)(29,117,65,81)(30,116,66,80)(31,115,67,79)(32,114,68,78)(33,113,69,77)(34,112,70,76)(35,111,71,75)(36,110,72,74)>;
G:=Group( (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), (1,73,37,109)(2,144,38,108)(3,143,39,107)(4,142,40,106)(5,141,41,105)(6,140,42,104)(7,139,43,103)(8,138,44,102)(9,137,45,101)(10,136,46,100)(11,135,47,99)(12,134,48,98)(13,133,49,97)(14,132,50,96)(15,131,51,95)(16,130,52,94)(17,129,53,93)(18,128,54,92)(19,127,55,91)(20,126,56,90)(21,125,57,89)(22,124,58,88)(23,123,59,87)(24,122,60,86)(25,121,61,85)(26,120,62,84)(27,119,63,83)(28,118,64,82)(29,117,65,81)(30,116,66,80)(31,115,67,79)(32,114,68,78)(33,113,69,77)(34,112,70,76)(35,111,71,75)(36,110,72,74) );
G=PermutationGroup([[(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)], [(1,73,37,109),(2,144,38,108),(3,143,39,107),(4,142,40,106),(5,141,41,105),(6,140,42,104),(7,139,43,103),(8,138,44,102),(9,137,45,101),(10,136,46,100),(11,135,47,99),(12,134,48,98),(13,133,49,97),(14,132,50,96),(15,131,51,95),(16,130,52,94),(17,129,53,93),(18,128,54,92),(19,127,55,91),(20,126,56,90),(21,125,57,89),(22,124,58,88),(23,123,59,87),(24,122,60,86),(25,121,61,85),(26,120,62,84),(27,119,63,83),(28,118,64,82),(29,117,65,81),(30,116,66,80),(31,115,67,79),(32,114,68,78),(33,113,69,77),(34,112,70,76),(35,111,71,75),(36,110,72,74)]])
Dic36 is a maximal subgroup of
C144⋊C2 Dic72 D8.D9 C9⋊Q32 D72⋊7C2 C8.D18 D8⋊3D9 SD16⋊D9 Q16×D9 Dic108 C3⋊Dic36 C72.C6 C24.D9
Dic36 is a maximal quotient of
C36.45D4 C72⋊1C4 Dic108 C3⋊Dic36 C24.D9
39 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | Q16 | D9 | D12 | D18 | Dic12 | D36 | Dic36 |
| kernel | Dic36 | C72 | Dic18 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 12 |
Matrix representation of Dic36 ►in GL2(𝔽73) generated by
| 22 | 32 |
| 41 | 63 |
| 37 | 62 |
| 25 | 36 |
G:=sub<GL(2,GF(73))| [22,41,32,63],[37,25,62,36] >;
Dic36 in GAP, Magma, Sage, TeX
{\rm Dic}_{36} % in TeX
G:=Group("Dic36"); // GroupNames label
G:=SmallGroup(144,4);
// by ID
G=gap.SmallGroup(144,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b|a^72=1,b^2=a^36,b*a*b^-1=a^-1>;
// generators/relations
Export