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G = Dic36order 144 = 24·32

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic36, C8.D9, C91Q16, 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

C1C36 — Dic36
C1C3C9C18C36Dic18 — Dic36
C9C18C36 — Dic36
C1C2C4C8

Generators and relations for Dic36
 G = < a,b | a72=1, b2=a36, bab-1=a-1 >

18C4
18C4
9Q8
9Q8
6Dic3
6Dic3
9Q16
3Dic6
3Dic6
2Dic9
2Dic9
3Dic12

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

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

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

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

Dic36 is a maximal subgroup of
C144⋊C2  Dic72  D8.D9  C9⋊Q32  D727C2  C8.D18  D83D9  SD16⋊D9  Q16×D9  Dic108  C3⋊Dic36  C72.C6  C24.D9
Dic36 is a maximal quotient of
C36.45D4  C721C4  Dic108  C3⋊Dic36  C24.D9

39 conjugacy classes

class 1  2  3 4A4B4C 6 8A8B9A9B9C12A12B18A18B18C24A24B24C24D36A···36F72A···72L
order12344468899912121818182424242436···3672···72
size112236362222222222222222···22···2

39 irreducible representations

dim1112222222222
type++++++-+++-+-
imageC1C2C2S3D4D6Q16D9D12D18Dic12D36Dic36
kernelDic36C72Dic18C24C18C12C9C8C6C4C3C2C1
# reps11211123234612

Matrix representation of Dic36 in GL2(𝔽73) generated by

2232
4163
,
3762
2536
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

Subgroup lattice of Dic36 in TeX

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