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## G = C12×Dic9order 432 = 24·33

### Direct product of C12 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C12×Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C6×C18 — C6×Dic9 — C12×Dic9
 Lower central C9 — C12×Dic9
 Upper central C1 — C2×C12

Generators and relations for C12×Dic9
G = < a,b,c | a12=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 262 in 106 conjugacy classes, 62 normal (26 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, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C3×C18, C3×C18, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C3×Dic9, C3×C36, C6×C18, C4×Dic9, Dic3×C12, C6×Dic9, C6×C36, C12×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, D9, C3×S3, C4×S3, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C3×D9, C4×D9, C2×Dic9, S3×C12, C6×Dic3, C3×Dic9, C6×D9, C4×Dic9, Dic3×C12, C12×D9, C6×Dic9, C12×Dic9

Smallest permutation representation of C12×Dic9
On 144 points
Generators in S144
(1 39 76 140 13 51 88 134 7 45 82 128)(2 40 77 141 14 52 89 135 8 46 83 129)(3 41 78 142 15 53 90 136 9 47 84 130)(4 42 79 143 16 54 73 137 10 48 85 131)(5 43 80 144 17 37 74 138 11 49 86 132)(6 44 81 127 18 38 75 139 12 50 87 133)(19 112 106 68 25 118 94 56 31 124 100 62)(20 113 107 69 26 119 95 57 32 125 101 63)(21 114 108 70 27 120 96 58 33 126 102 64)(22 115 91 71 28 121 97 59 34 109 103 65)(23 116 92 72 29 122 98 60 35 110 104 66)(24 117 93 55 30 123 99 61 36 111 105 67)
(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 26 10 35)(2 25 11 34)(3 24 12 33)(4 23 13 32)(5 22 14 31)(6 21 15 30)(7 20 16 29)(8 19 17 28)(9 36 18 27)(37 121 46 112)(38 120 47 111)(39 119 48 110)(40 118 49 109)(41 117 50 126)(42 116 51 125)(43 115 52 124)(44 114 53 123)(45 113 54 122)(55 133 64 142)(56 132 65 141)(57 131 66 140)(58 130 67 139)(59 129 68 138)(60 128 69 137)(61 127 70 136)(62 144 71 135)(63 143 72 134)(73 98 82 107)(74 97 83 106)(75 96 84 105)(76 95 85 104)(77 94 86 103)(78 93 87 102)(79 92 88 101)(80 91 89 100)(81 108 90 99)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G ··· 6O 9A ··· 9I 12A ··· 12H 12I ··· 12T 12U ··· 12AJ 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 2 2 2 1 1 1 1 9 ··· 9 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 D9 C3×S3 C4×S3 Dic9 C3×Dic3 D18 S3×C6 C3×D9 C4×D9 S3×C12 C3×Dic9 C6×D9 C12×D9 kernel C12×Dic9 C6×Dic9 C6×C36 C4×Dic9 C3×Dic9 C3×C36 C2×Dic9 C2×C36 Dic9 C36 C6×C12 C3×C12 C62 C2×C12 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 1 2 1 3 2 4 6 4 3 2 6 12 8 12 6 24

Matrix representation of C12×Dic9 in GL4(𝔽37) generated by

 8 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 36 0 0 0 0 16 0 0 0 0 7
,
 1 0 0 0 0 31 0 0 0 0 0 36 0 0 36 0
G:=sub<GL(4,GF(37))| [8,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,36,0,0,0,0,16,0,0,0,0,7],[1,0,0,0,0,31,0,0,0,0,0,36,0,0,36,0] >;

C12×Dic9 in GAP, Magma, Sage, TeX

C_{12}\times {\rm Dic}_9
% in TeX

G:=Group("C12xDic9");
// GroupNames label

G:=SmallGroup(432,128);
// by ID

G=gap.SmallGroup(432,128);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^12=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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