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## G = C9×Dic12order 432 = 24·33

### Direct product of C9 and Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×Dic12
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×Dic6 — C9×Dic12
 Lower central C3 — C6 — C12 — C9×Dic12
 Upper central C1 — C18 — C36 — C72

Generators and relations for C9×Dic12
G = < a,b,c | a9=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 124 in 62 conjugacy classes, 33 normal (27 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, C36, C36, C3×Dic3, C3×C12, Dic12, C3×Q16, C3×C18, C72, C72, Q8×C9, C3×C24, C3×Dic6, C9×Dic3, C3×C36, C9×Q16, C3×Dic12, C3×C72, C9×Dic6, C9×Dic12
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, Q16, C18, C3×S3, D12, C3×D4, C2×C18, S3×C6, Dic12, C3×Q16, S3×C9, D4×C9, C3×D12, S3×C18, C9×Q16, C3×Dic12, C9×D12, C9×Dic12

Smallest permutation representation of C9×Dic12
On 144 points
Generators in S144
(1 126 37 9 134 45 17 142 29)(2 127 38 10 135 46 18 143 30)(3 128 39 11 136 47 19 144 31)(4 129 40 12 137 48 20 121 32)(5 130 41 13 138 25 21 122 33)(6 131 42 14 139 26 22 123 34)(7 132 43 15 140 27 23 124 35)(8 133 44 16 141 28 24 125 36)(49 87 120 65 79 112 57 95 104)(50 88 97 66 80 113 58 96 105)(51 89 98 67 81 114 59 73 106)(52 90 99 68 82 115 60 74 107)(53 91 100 69 83 116 61 75 108)(54 92 101 70 84 117 62 76 109)(55 93 102 71 85 118 63 77 110)(56 94 103 72 86 119 64 78 111)
(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 65 13 53)(2 64 14 52)(3 63 15 51)(4 62 16 50)(5 61 17 49)(6 60 18 72)(7 59 19 71)(8 58 20 70)(9 57 21 69)(10 56 22 68)(11 55 23 67)(12 54 24 66)(25 100 37 112)(26 99 38 111)(27 98 39 110)(28 97 40 109)(29 120 41 108)(30 119 42 107)(31 118 43 106)(32 117 44 105)(33 116 45 104)(34 115 46 103)(35 114 47 102)(36 113 48 101)(73 144 85 132)(74 143 86 131)(75 142 87 130)(76 141 88 129)(77 140 89 128)(78 139 90 127)(79 138 91 126)(80 137 92 125)(81 136 93 124)(82 135 94 123)(83 134 95 122)(84 133 96 121)

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

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

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

135 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 9A ··· 9F 9G ··· 9L 12A ··· 12H 12I 12J 12K 12L 18A ··· 18F 18G ··· 18L 24A ··· 24P 36A ··· 36R 36S ··· 36AD 72A ··· 72AJ order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 9 ··· 9 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 1 1 2 2 2 2 12 12 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 12 12 12 12 1 ··· 1 2 ··· 2 2 ··· 2 2 ··· 2 12 ··· 12 2 ··· 2

135 irreducible representations

 dim 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 2 2 type + + + + + + - + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D4 D6 Q16 C3×S3 D12 C3×D4 S3×C6 Dic12 C3×Q16 S3×C9 D4×C9 C3×D12 S3×C18 C9×Q16 C3×Dic12 C9×D12 C9×Dic12 kernel C9×Dic12 C3×C72 C9×Dic6 C3×Dic12 C3×C24 C3×Dic6 Dic12 C24 Dic6 C72 C3×C18 C36 C3×C9 C24 C18 C3×C6 C12 C9 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 6 6 12 1 1 1 2 2 2 2 2 4 4 6 6 4 6 12 8 12 24

Matrix representation of C9×Dic12 in GL2(𝔽73) generated by

 2 0 0 2
,
 56 0 0 30
,
 0 1 72 0
G:=sub<GL(2,GF(73))| [2,0,0,2],[56,0,0,30],[0,72,1,0] >;

C9×Dic12 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_{12}
% in TeX

G:=Group("C9xDic12");
// GroupNames label

G:=SmallGroup(432,113);
// by ID

G=gap.SmallGroup(432,113);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,596,142,2355,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^9=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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