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## G = Dic3×C36order 432 = 24·33

### Direct product of C36 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C36
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C18 — Dic3×C18 — Dic3×C36
 Lower central C3 — Dic3×C36
 Upper central C1 — C2×C36

Generators and relations for Dic3×C36
G = < a,b,c | a36=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 164 in 106 conjugacy classes, 69 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C3×C18, C3×C18, C2×C36, C2×C36, C6×Dic3, C6×C12, C9×Dic3, C3×C36, C6×C18, C4×C36, Dic3×C12, Dic3×C18, C6×C36, Dic3×C36
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, Dic3, C12, D6, C2×C6, C42, C18, C3×S3, C4×S3, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, C4×Dic3, C4×C12, S3×C9, C2×C36, S3×C12, C6×Dic3, C9×Dic3, S3×C18, C4×C36, Dic3×C12, S3×C36, Dic3×C18, Dic3×C36

Smallest permutation representation of Dic3×C36
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 54 13 66 25 42)(2 55 14 67 26 43)(3 56 15 68 27 44)(4 57 16 69 28 45)(5 58 17 70 29 46)(6 59 18 71 30 47)(7 60 19 72 31 48)(8 61 20 37 32 49)(9 62 21 38 33 50)(10 63 22 39 34 51)(11 64 23 40 35 52)(12 65 24 41 36 53)(73 113 97 137 85 125)(74 114 98 138 86 126)(75 115 99 139 87 127)(76 116 100 140 88 128)(77 117 101 141 89 129)(78 118 102 142 90 130)(79 119 103 143 91 131)(80 120 104 144 92 132)(81 121 105 109 93 133)(82 122 106 110 94 134)(83 123 107 111 95 135)(84 124 108 112 96 136)
(1 143 66 79)(2 144 67 80)(3 109 68 81)(4 110 69 82)(5 111 70 83)(6 112 71 84)(7 113 72 85)(8 114 37 86)(9 115 38 87)(10 116 39 88)(11 117 40 89)(12 118 41 90)(13 119 42 91)(14 120 43 92)(15 121 44 93)(16 122 45 94)(17 123 46 95)(18 124 47 96)(19 125 48 97)(20 126 49 98)(21 127 50 99)(22 128 51 100)(23 129 52 101)(24 130 53 102)(25 131 54 103)(26 132 55 104)(27 133 56 105)(28 134 57 106)(29 135 58 107)(30 136 59 108)(31 137 60 73)(32 138 61 74)(33 139 62 75)(34 140 63 76)(35 141 64 77)(36 142 65 78)

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

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

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

216 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G ··· 6O 9A ··· 9F 9G ··· 9L 12A ··· 12H 12I ··· 12T 12U ··· 12AJ 18A ··· 18R 18S ··· 18AJ 36A ··· 36X 36Y ··· 36AV 36AW ··· 36CR order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 1 1 1 1 2 2 2 1 1 1 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C9 C12 C12 C18 C18 C36 C36 S3 Dic3 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C9 S3×C12 C9×Dic3 S3×C18 S3×C36 kernel Dic3×C36 Dic3×C18 C6×C36 Dic3×C12 C9×Dic3 C3×C36 C6×Dic3 C6×C12 C4×Dic3 C3×Dic3 C3×C12 C2×Dic3 C2×C12 Dic3 C12 C2×C36 C36 C2×C18 C2×C12 C18 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 6 16 8 12 6 48 24 1 2 1 2 4 4 2 6 8 12 6 24

Matrix representation of Dic3×C36 in GL3(𝔽37) generated by

 17 0 0 0 2 0 0 0 2
,
 1 0 0 0 27 0 0 0 11
,
 36 0 0 0 0 31 0 31 0
G:=sub<GL(3,GF(37))| [17,0,0,0,2,0,0,0,2],[1,0,0,0,27,0,0,0,11],[36,0,0,0,0,31,0,31,0] >;

Dic3×C36 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{36}
% in TeX

G:=Group("Dic3xC36");
// GroupNames label

G:=SmallGroup(432,131);
// by ID

G=gap.SmallGroup(432,131);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,268,14118]);
// Polycyclic

G:=Group<a,b,c|a^36=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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