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G = C12xDic6order 288 = 25·32

Direct product of C12 and Dic6

direct product, metabelian, supersoluble, monomial

Aliases: C12xDic6, C122.7C2, C62.159C23, C3:1(Q8xC12), C12:3(C3xQ8), C6.1(C6xQ8), C32:9(C4xQ8), C4.9(S3xC12), (C3xC12):10Q8, C12.87(C4xS3), (C4xC12).11C6, (C4xC12).23S3, C42.3(C3xS3), C2.1(C6xDic6), C12.19(C2xC12), (C2xC12).436D6, Dic3:C4.7C6, C4:Dic3.13C6, C6.1(C22xC12), (C4xDic3).7C6, C6.47(C2xDic6), (C6xDic6).20C2, (C2xDic6).10C6, Dic3.1(C2xC12), C6.110(C4oD12), (C6xC12).278C22, (Dic3xC12).18C2, (C6xDic3).157C22, C2.4(S3xC2xC12), C6.100(S3xC2xC4), C6.1(C3xC4oD4), C22.8(S3xC2xC6), (C2xC4).95(S3xC6), C2.1(C3xC4oD12), (C3xC6).44(C2xQ8), (C3xC12).99(C2xC4), (C2xC12).125(C2xC6), (C3xC6).91(C4oD4), (C3xC4:Dic3).27C2, (C3xC6).72(C22xC4), (C2xC6).14(C22xC6), (C3xDic3:C4).16C2, (C2xC6).292(C22xS3), (C3xDic3).17(C2xC4), (C2xDic3).15(C2xC6), SmallGroup(288,639)

Series: Derived Chief Lower central Upper central

C1C6 — C12xDic6
C1C3C6C2xC6C62C6xDic3C6xDic6 — C12xDic6
C3C6 — C12xDic6
C1C2xC12C4xC12

Generators and relations for C12xDic6
 G = < a,b,c | a12=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 274 in 155 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2xC4, C2xC4, Q8, C32, Dic3, Dic3, C12, C12, C2xC6, C2xC6, C42, C42, C4:C4, C2xQ8, C3xC6, Dic6, C2xDic3, C2xC12, C2xC12, C3xQ8, C4xQ8, C3xDic3, C3xDic3, C3xC12, C3xC12, C62, C4xDic3, Dic3:C4, C4:Dic3, C4xC12, C4xC12, C3xC4:C4, C2xDic6, C6xQ8, C3xDic6, C6xDic3, C6xC12, C4xDic6, Q8xC12, Dic3xC12, C3xDic3:C4, C3xC4:Dic3, C122, C6xDic6, C12xDic6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, Q8, C23, C12, D6, C2xC6, C22xC4, C2xQ8, C4oD4, C3xS3, Dic6, C4xS3, C2xC12, C3xQ8, C22xS3, C22xC6, C4xQ8, S3xC6, C2xDic6, S3xC2xC4, C4oD12, C22xC12, C6xQ8, C3xC4oD4, C3xDic6, S3xC12, S3xC2xC6, C4xDic6, Q8xC12, C6xDic6, S3xC2xC12, C3xC4oD12, C12xDic6

Smallest permutation representation of C12xDic6
On 96 points
Generators in S96
(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)
(1 89 42 76 9 85 38 84 5 93 46 80)(2 90 43 77 10 86 39 73 6 94 47 81)(3 91 44 78 11 87 40 74 7 95 48 82)(4 92 45 79 12 88 41 75 8 96 37 83)(13 25 55 72 17 29 59 64 21 33 51 68)(14 26 56 61 18 30 60 65 22 34 52 69)(15 27 57 62 19 31 49 66 23 35 53 70)(16 28 58 63 20 32 50 67 24 36 54 71)
(1 16 38 50)(2 17 39 51)(3 18 40 52)(4 19 41 53)(5 20 42 54)(6 21 43 55)(7 22 44 56)(8 23 45 57)(9 24 46 58)(10 13 47 59)(11 14 48 60)(12 15 37 49)(25 94 64 77)(26 95 65 78)(27 96 66 79)(28 85 67 80)(29 86 68 81)(30 87 69 82)(31 88 70 83)(32 89 71 84)(33 90 72 73)(34 91 61 74)(35 92 62 75)(36 93 63 76)

G:=sub<Sym(96)| (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), (1,89,42,76,9,85,38,84,5,93,46,80)(2,90,43,77,10,86,39,73,6,94,47,81)(3,91,44,78,11,87,40,74,7,95,48,82)(4,92,45,79,12,88,41,75,8,96,37,83)(13,25,55,72,17,29,59,64,21,33,51,68)(14,26,56,61,18,30,60,65,22,34,52,69)(15,27,57,62,19,31,49,66,23,35,53,70)(16,28,58,63,20,32,50,67,24,36,54,71), (1,16,38,50)(2,17,39,51)(3,18,40,52)(4,19,41,53)(5,20,42,54)(6,21,43,55)(7,22,44,56)(8,23,45,57)(9,24,46,58)(10,13,47,59)(11,14,48,60)(12,15,37,49)(25,94,64,77)(26,95,65,78)(27,96,66,79)(28,85,67,80)(29,86,68,81)(30,87,69,82)(31,88,70,83)(32,89,71,84)(33,90,72,73)(34,91,61,74)(35,92,62,75)(36,93,63,76)>;

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), (1,89,42,76,9,85,38,84,5,93,46,80)(2,90,43,77,10,86,39,73,6,94,47,81)(3,91,44,78,11,87,40,74,7,95,48,82)(4,92,45,79,12,88,41,75,8,96,37,83)(13,25,55,72,17,29,59,64,21,33,51,68)(14,26,56,61,18,30,60,65,22,34,52,69)(15,27,57,62,19,31,49,66,23,35,53,70)(16,28,58,63,20,32,50,67,24,36,54,71), (1,16,38,50)(2,17,39,51)(3,18,40,52)(4,19,41,53)(5,20,42,54)(6,21,43,55)(7,22,44,56)(8,23,45,57)(9,24,46,58)(10,13,47,59)(11,14,48,60)(12,15,37,49)(25,94,64,77)(26,95,65,78)(27,96,66,79)(28,85,67,80)(29,86,68,81)(30,87,69,82)(31,88,70,83)(32,89,71,84)(33,90,72,73)(34,91,61,74)(35,92,62,75)(36,93,63,76) );

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)], [(1,89,42,76,9,85,38,84,5,93,46,80),(2,90,43,77,10,86,39,73,6,94,47,81),(3,91,44,78,11,87,40,74,7,95,48,82),(4,92,45,79,12,88,41,75,8,96,37,83),(13,25,55,72,17,29,59,64,21,33,51,68),(14,26,56,61,18,30,60,65,22,34,52,69),(15,27,57,62,19,31,49,66,23,35,53,70),(16,28,58,63,20,32,50,67,24,36,54,71)], [(1,16,38,50),(2,17,39,51),(3,18,40,52),(4,19,41,53),(5,20,42,54),(6,21,43,55),(7,22,44,56),(8,23,45,57),(9,24,46,58),(10,13,47,59),(11,14,48,60),(12,15,37,49),(25,94,64,77),(26,95,65,78),(27,96,66,79),(28,85,67,80),(29,86,68,81),(30,87,69,82),(31,88,70,83),(32,89,71,84),(33,90,72,73),(34,91,61,74),(35,92,62,75),(36,93,63,76)]])

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H4I···4P6A···6F6G···6O12A···12H12I···12AZ12BA···12BP
order122233333444444444···46···66···612···1212···1212···12
size111111222111122226···61···12···21···12···26···6

108 irreducible representations

dim1111111111111122222222222222
type+++++++-+-
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12S3Q8D6C4oD4C3xS3Dic6C4xS3C3xQ8S3xC6C4oD12C3xC4oD4C3xDic6S3xC12C3xC4oD12
kernelC12xDic6Dic3xC12C3xDic3:C4C3xC4:Dic3C122C6xDic6C4xDic6C3xDic6C4xDic3Dic3:C4C4:Dic3C4xC12C2xDic6Dic6C4xC12C3xC12C2xC12C3xC6C42C12C12C12C2xC4C6C6C4C4C2
# reps12211128442221612322444644888

Matrix representation of C12xDic6 in GL3(F13) generated by

700
040
004
,
100
020
007
,
1200
001
0120
G:=sub<GL(3,GF(13))| [7,0,0,0,4,0,0,0,4],[1,0,0,0,2,0,0,0,7],[12,0,0,0,0,12,0,1,0] >;

C12xDic6 in GAP, Magma, Sage, TeX

C_{12}\times {\rm Dic}_6
% in TeX

G:=Group("C12xDic6");
// GroupNames label

G:=SmallGroup(288,639);
// by ID

G=gap.SmallGroup(288,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,142,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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