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G = C12×Dic9order 432 = 24·33

Direct product of C12 and Dic9

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C12×Dic9, C366C12, C62.118D6, C93(C4×C12), (C3×C36)⋊4C4, (C3×C9)⋊3C42, C2.2(C12×D9), C6.23(C4×D9), C6.11(S3×C12), (C6×C12).47S3, (C2×C36).20C6, (C6×C36).14C2, (C2×C12).22D9, (C2×C6).46D18, C6.8(C6×Dic3), C2.2(C6×Dic9), C22.3(C6×D9), C18.17(C2×C12), (C6×Dic9).9C2, (C2×Dic9).8C6, C6.19(C2×Dic9), C3.1(Dic3×C12), (C6×C18).32C22, C12.12(C3×Dic3), (C3×C12).24Dic3, C32.3(C4×Dic3), (C2×C4).6(C3×D9), (C2×C6).34(S3×C6), (C3×C6).65(C4×S3), (C2×C12).30(C3×S3), (C3×C18).20(C2×C4), (C2×C18).23(C2×C6), (C3×C6).56(C2×Dic3), SmallGroup(432,128)

Series: Derived Chief Lower central Upper central

C1C9 — C12×Dic9
C1C3C9C18C2×C18C6×C18C6×Dic9 — C12×Dic9
C9 — C12×Dic9
C1C2×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 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], C9, C9, C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×C9, Dic9 [×4], C36 [×2], C36 [×2], C2×C18, C2×C18, C3×Dic3 [×4], C3×C12 [×2], C62, C4×Dic3, C4×C12, C3×C18, C3×C18 [×2], C2×Dic9 [×2], C2×C36, C2×C36, C6×Dic3 [×2], C6×C12, C3×Dic9 [×4], C3×C36 [×2], C6×C18, C4×Dic9, Dic3×C12, C6×Dic9 [×2], C6×C36, C12×Dic9
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, D9, C3×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×3], Dic9 [×2], D18, C3×Dic3 [×2], S3×C6, C4×Dic3, C4×C12, C3×D9, C4×D9 [×2], C2×Dic9, S3×C12 [×2], C6×Dic3, C3×Dic9 [×2], C6×D9, C4×Dic9, Dic3×C12, C12×D9 [×2], C6×Dic9, C12×Dic9

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

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

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

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

144 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E···4L6A···6F6G···6O9A···9I12A···12H12I···12T12U···12AJ18A···18AA36A···36AJ
order12223333344444···46···66···69···912···1212···1212···1218···1836···36
size11111122211119···91···12···22···21···12···29···92···22···2

144 irreducible representations

dim11111111112222222222222222
type++++-++-+
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6D9C3×S3C4×S3Dic9C3×Dic3D18S3×C6C3×D9C4×D9S3×C12C3×Dic9C6×D9C12×D9
kernelC12×Dic9C6×Dic9C6×C36C4×Dic9C3×Dic9C3×C36C2×Dic9C2×C36Dic9C36C6×C12C3×C12C62C2×C12C2×C12C3×C6C12C12C2×C6C2×C6C2×C4C6C6C4C22C2
# reps121284421681213246432612812624

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

8000
01100
00110
00011
,
1000
03600
00160
0007
,
1000
03100
00036
00360
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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