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G = C9×Dic12order 432 = 24·33

Direct product of C9 and Dic12

direct product, metacyclic, supersoluble, monomial

Aliases: C9×Dic12, C72.9S3, C24.1C18, C36.73D6, C18.21D12, Dic6.1C18, C8.(S3×C9), (C3×C9)⋊4Q16, C31(C9×Q16), C6.3(D4×C9), (C3×C72).6C2, C2.5(C9×D12), C4.10(S3×C18), C24.12(C3×S3), (C3×C24).13C6, C6.33(C3×D12), (C3×C18).21D4, (C3×Dic12).C3, C12.108(S3×C6), C12.10(C2×C18), (C9×Dic6).3C2, (C3×Dic6).9C6, C3.4(C3×Dic12), C32.2(C3×Q16), (C3×C36).75C22, (C3×C6).41(C3×D4), (C3×C12).79(C2×C6), SmallGroup(432,113)

Series: Derived Chief Lower central Upper central

C1C12 — C9×Dic12
C1C3C6C3×C6C3×C12C3×C36C9×Dic6 — C9×Dic12
C3C6C12 — C9×Dic12
C1C18C36C72

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 [×2], C3, C4, C4 [×2], C6 [×2], C6, C8, Q8 [×2], C9, C9, C32, Dic3 [×2], C12 [×2], C12 [×3], Q16, C18, C18, C3×C6, C24 [×2], C24, Dic6 [×2], C3×Q8 [×2], C3×C9, C36, C36 [×3], C3×Dic3 [×2], C3×C12, Dic12, C3×Q16, C3×C18, C72, C72, Q8×C9 [×2], C3×C24, C3×Dic6 [×2], C9×Dic3 [×2], C3×C36, C9×Q16, C3×Dic12, C3×C72, C9×Dic6 [×2], C9×Dic12
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, C9, D6, C2×C6, Q16, C18 [×3], 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
(1 127 86 9 135 94 17 143 78)(2 128 87 10 136 95 18 144 79)(3 129 88 11 137 96 19 121 80)(4 130 89 12 138 73 20 122 81)(5 131 90 13 139 74 21 123 82)(6 132 91 14 140 75 22 124 83)(7 133 92 15 141 76 23 125 84)(8 134 93 16 142 77 24 126 85)(25 71 116 41 63 108 33 55 100)(26 72 117 42 64 109 34 56 101)(27 49 118 43 65 110 35 57 102)(28 50 119 44 66 111 36 58 103)(29 51 120 45 67 112 37 59 104)(30 52 97 46 68 113 38 60 105)(31 53 98 47 69 114 39 61 106)(32 54 99 48 70 115 40 62 107)
(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 39 13 27)(2 38 14 26)(3 37 15 25)(4 36 16 48)(5 35 17 47)(6 34 18 46)(7 33 19 45)(8 32 20 44)(9 31 21 43)(10 30 22 42)(11 29 23 41)(12 28 24 40)(49 127 61 139)(50 126 62 138)(51 125 63 137)(52 124 64 136)(53 123 65 135)(54 122 66 134)(55 121 67 133)(56 144 68 132)(57 143 69 131)(58 142 70 130)(59 141 71 129)(60 140 72 128)(73 119 85 107)(74 118 86 106)(75 117 87 105)(76 116 88 104)(77 115 89 103)(78 114 90 102)(79 113 91 101)(80 112 92 100)(81 111 93 99)(82 110 94 98)(83 109 95 97)(84 108 96 120)

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

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

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

135 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B9A···9F9G···9L12A···12H12I12J12K12L18A···18F18G···18L24A···24P36A···36R36S···36AD72A···72AJ
order123333344466666889···99···912···121212121218···1818···1824···2436···3636···3672···72
size11112222121211222221···12···22···2121212121···12···22···22···212···122···2

135 irreducible representations

dim111111111222222222222222222
type++++++-+-
imageC1C2C2C3C6C6C9C18C18S3D4D6Q16C3×S3D12C3×D4S3×C6Dic12C3×Q16S3×C9D4×C9C3×D12S3×C18C9×Q16C3×Dic12C9×D12C9×Dic12
kernelC9×Dic12C3×C72C9×Dic6C3×Dic12C3×C24C3×Dic6Dic12C24Dic6C72C3×C18C36C3×C9C24C18C3×C6C12C9C32C8C6C6C4C3C3C2C1
# reps1122246612111222224466461281224

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

20
02
,
560
030
,
01
720
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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