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G = C6.Dic12order 288 = 25·32

1st non-split extension by C6 of Dic12 acting via Dic12/Dic6=C2

metabelian, supersoluble, monomial

Aliases: C6.6Dic12, C62.21D4, Dic61Dic3, (C3×C6).4Q16, C12.17(C4×S3), (C3×Dic6)⋊2C4, (C3×C12).36D4, (C2×C6).51D12, C4.2(S3×Dic3), C6.9(C24⋊C2), C6.37(D6⋊C4), (C2×C12).274D6, (C6×Dic6).3C2, (C2×Dic6).1S3, (C3×C6).11SD16, C6.1(C3⋊Q16), C31(Q82Dic3), C12.32(C3⋊D4), C2.8(D6⋊Dic3), (C6×C12).28C22, C6.2(Q82S3), C33(C2.Dic12), C12.23(C2×Dic3), C325(Q8⋊C4), C4.12(D6⋊S3), C12⋊Dic3.9C2, C6.7(C6.D4), C2.1(C323Q16), C2.2(C325SD16), C22.14(C3⋊D12), (C6×C3⋊C8).1C2, (C2×C4).54S32, (C2×C3⋊C8).1S3, (C3×C12).26(C2×C4), (C2×C6).32(C3⋊D4), (C3×C6).33(C22⋊C4), SmallGroup(288,214)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C6.Dic12
C1C3C32C3×C6C3×C12C6×C12C6×Dic6 — C6.Dic12
C32C3×C6C3×C12 — C6.Dic12
C1C22C2×C4

Generators and relations for C6.Dic12
 G = < a,b,c | a6=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 330 in 97 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×6], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8, C24, Dic6 [×2], Dic6, C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C3×Q8 [×3], Q8⋊C4, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3 [×3], C2×C24, C2×Dic6, C6×Q8, C3×C3⋊C8, C3×Dic6 [×2], C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C2.Dic12, Q82Dic3, C6×C3⋊C8, C12⋊Dic3, C6×Dic6, C6.Dic12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, SD16, Q16, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], Q8⋊C4, S32, C24⋊C2, Dic12, D6⋊C4, Q82S3, C3⋊Q16, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C2.Dic12, Q82Dic3, C325SD16, C323Q16, D6⋊Dic3, C6.Dic12

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

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

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A8B8C8D12A12B12C12D12E···12J12K12L12M12N24A···24H
order12223334444446···666688881212121212···121212121224···24
size111122422121236362···2444666622224···4121212126···6

48 irreducible representations

dim111112222222222222244444444
type++++++++-+-+-++---++-
imageC1C2C2C2C4S3S3D4D4Dic3D6SD16Q16C4×S3C3⋊D4D12C3⋊D4C24⋊C2Dic12S32Q82S3C3⋊Q16S3×Dic3D6⋊S3C3⋊D12C325SD16C323Q16
kernelC6.Dic12C6×C3⋊C8C12⋊Dic3C6×Dic6C3×Dic6C2×C3⋊C8C2×Dic6C3×C12C62Dic6C2×C12C3×C6C3×C6C12C12C2×C6C2×C6C6C6C2×C4C6C6C4C4C22C2C2
# reps111141111222224224411111122

Matrix representation of C6.Dic12 in GL6(𝔽73)

7200000
0720000
0017200
001000
000010
000001
,
59550000
8260000
0002700
0027000
0000667
00006659
,
24250000
47490000
001000
000100
0000714
0000766

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,8,0,0,0,0,55,26,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,66,66,0,0,0,0,7,59],[24,47,0,0,0,0,25,49,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,14,66] >;

C6.Dic12 in GAP, Magma, Sage, TeX

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

G:=Group("C6.Dic12");
// GroupNames label

G:=SmallGroup(288,214);
// by ID

G=gap.SmallGroup(288,214);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

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