Copied to
clipboard

G = C6×Dic12order 288 = 25·32

Direct product of C6 and Dic12

direct product, metabelian, supersoluble, monomial

Aliases: C6×Dic12, C24.83D6, C12.69D12, C62.88D4, C61(C3×Q16), C31(C6×Q16), (C3×C6)⋊4Q16, C8.16(S3×C6), C6.12(C6×D4), C4.8(C3×D12), C329(C2×Q16), (C2×C24).10C6, C24.20(C2×C6), (C2×C24).21S3, (C6×C24).12C2, C12.31(C3×D4), (C2×C6).75D12, C2.14(C6×D12), C6.100(C2×D12), (C2×C12).444D6, (C3×C12).133D4, (C2×Dic6).4C6, Dic6.7(C2×C6), C12.31(C22×C6), (C3×C24).58C22, (C6×Dic6).19C2, C22.14(C3×D12), C12.218(C22×S3), (C3×C12).163C23, (C6×C12).323C22, (C3×Dic6).47C22, C4.29(S3×C2×C6), (C2×C8).4(C3×S3), (C2×C4).81(S3×C6), (C2×C6).23(C3×D4), (C3×C6).182(C2×D4), (C2×C12).108(C2×C6), SmallGroup(288,676)

Series: Derived Chief Lower central Upper central

C1C12 — C6×Dic12
C1C3C6C12C3×C12C3×Dic6C6×Dic6 — C6×Dic12
C3C6C12 — C6×Dic12
C1C2×C6C2×C12C2×C24

Generators and relations for C6×Dic12
 G = < a,b,c | a6=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 282 in 131 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], Dic6 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×6], C2×Q16, C3×Dic3 [×4], C3×C12 [×2], C62, Dic12 [×4], C2×C24 [×2], C2×C24, C3×Q16 [×4], C2×Dic6 [×2], C6×Q8 [×2], C3×C24 [×2], C3×Dic6 [×4], C3×Dic6 [×2], C6×Dic3 [×2], C6×C12, C2×Dic12, C6×Q16, C3×Dic12 [×4], C6×C24, C6×Dic6 [×2], C6×Dic12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], Q16 [×2], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×Q16, S3×C6 [×3], Dic12 [×2], C3×Q16 [×2], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C2×Dic12, C6×Q16, C3×Dic12 [×2], C6×D12, C6×Dic12

Smallest permutation representation of C6×Dic12
On 96 points
Generators in S96
(1 85 9 93 17 77)(2 86 10 94 18 78)(3 87 11 95 19 79)(4 88 12 96 20 80)(5 89 13 73 21 81)(6 90 14 74 22 82)(7 91 15 75 23 83)(8 92 16 76 24 84)(25 49 41 65 33 57)(26 50 42 66 34 58)(27 51 43 67 35 59)(28 52 44 68 36 60)(29 53 45 69 37 61)(30 54 46 70 38 62)(31 55 47 71 39 63)(32 56 48 72 40 64)
(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 41 13 29)(2 40 14 28)(3 39 15 27)(4 38 16 26)(5 37 17 25)(6 36 18 48)(7 35 19 47)(8 34 20 46)(9 33 21 45)(10 32 22 44)(11 31 23 43)(12 30 24 42)(49 89 61 77)(50 88 62 76)(51 87 63 75)(52 86 64 74)(53 85 65 73)(54 84 66 96)(55 83 67 95)(56 82 68 94)(57 81 69 93)(58 80 70 92)(59 79 71 91)(60 78 72 90)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O8A8B8C8D12A···12P12Q···12X24A···24AF
order1222333334444446···66···6888812···1212···1224···24
size11111122222121212121···12···222222···212···122···2

90 irreducible representations

dim11111111222222222222222222
type+++++++++-++-
imageC1C2C2C2C3C6C6C6S3D4D4D6D6Q16C3×S3D12C3×D4D12C3×D4S3×C6S3×C6Dic12C3×Q16C3×D12C3×D12C3×Dic12
kernelC6×Dic12C3×Dic12C6×C24C6×Dic6C2×Dic12Dic12C2×C24C2×Dic6C2×C24C3×C12C62C24C2×C12C3×C6C2×C8C12C12C2×C6C2×C6C8C2×C4C6C6C4C22C2
# reps141228241112142222242884416

Matrix representation of C6×Dic12 in GL4(𝔽73) generated by

65000
06500
00650
00065
,
70000
562400
00660
00052
,
463000
392700
00010
00510
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,65,0,0,0,0,65],[70,56,0,0,0,24,0,0,0,0,66,0,0,0,0,52],[46,39,0,0,30,27,0,0,0,0,0,51,0,0,10,0] >;

C6×Dic12 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_{12}
% in TeX

G:=Group("C6xDic12");
// GroupNames label

G:=SmallGroup(288,676);
// by ID

G=gap.SmallGroup(288,676);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,394,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽