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## G = C3×Dic36order 432 = 24·33

### Direct product of C3 and Dic36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×Dic36
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×Dic18 — C3×Dic36
 Lower central C9 — C18 — C36 — C3×Dic36
 Upper central C1 — C6 — C12 — C24

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

Smallest permutation representation of 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

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