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## G = C6.Dic12order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C6.Dic12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×Dic6 — C6.Dic12
 Lower central C32 — C3×C6 — C3×C12 — C6.Dic12
 Upper central C1 — C22 — C2×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 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 2 2 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + - + - + + - - - + + - image C1 C2 C2 C2 C4 S3 S3 D4 D4 Dic3 D6 SD16 Q16 C4×S3 C3⋊D4 D12 C3⋊D4 C24⋊C2 Dic12 S32 Q8⋊2S3 C3⋊Q16 S3×Dic3 D6⋊S3 C3⋊D12 C32⋊5SD16 C32⋊3Q16 kernel C6.Dic12 C6×C3⋊C8 C12⋊Dic3 C6×Dic6 C3×Dic6 C2×C3⋊C8 C2×Dic6 C3×C12 C62 Dic6 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C2×C6 C6 C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 2 2 4 4 1 1 1 1 1 1 2 2

Matrix representation of C6.Dic12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 59 55 0 0 0 0 8 26 0 0 0 0 0 0 0 27 0 0 0 0 27 0 0 0 0 0 0 0 66 7 0 0 0 0 66 59
,
 24 25 0 0 0 0 47 49 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 14 0 0 0 0 7 66

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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