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## G = Dic36order 144 = 24·32

### Dicyclic group

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

 Derived series C1 — C36 — Dic36
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — Dic36
 Lower central C9 — C18 — C36 — Dic36
 Upper central C1 — C2 — C4 — C8

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)
(1 123 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 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

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