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## G = C12×Dic6order 288 = 25·32

### Direct product of C12 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C6×Dic6 — C12×Dic6
 Lower central C3 — C6 — C12×Dic6
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C12×Dic6
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 [×3], C3 [×2], C3, C4 [×4], C4 [×7], C22, C6 [×6], C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C32, Dic3 [×4], Dic3 [×2], C12 [×8], C12 [×14], C2×C6 [×2], C2×C6, C42, C42 [×2], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×4], C2×Dic3 [×4], C2×C12 [×6], C2×C12 [×7], C3×Q8 [×4], C4×Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×4], C3×C12, C62, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12 [×2], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6, C6×Q8, C3×Dic6 [×4], C6×Dic3 [×4], C6×C12 [×3], C4×Dic6, Q8×C12, Dic3×C12 [×2], C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C122, C6×Dic6, C12×Dic6
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], Q8 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, C3×S3, Dic6 [×2], C4×S3 [×2], C2×C12 [×6], C3×Q8 [×2], C22×S3, C22×C6, C4×Q8, S3×C6 [×3], C2×Dic6, S3×C2×C4, C4○D12, C22×C12, C6×Q8, C3×C4○D4, C3×Dic6 [×2], S3×C12 [×2], S3×C2×C6, C4×Dic6, Q8×C12, C6×Dic6, S3×C2×C12, C3×C4○D12, C12×Dic6

Smallest permutation representation of C12×Dic6
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 52 86 13 9 60 94 21 5 56 90 17)(2 53 87 14 10 49 95 22 6 57 91 18)(3 54 88 15 11 50 96 23 7 58 92 19)(4 55 89 16 12 51 85 24 8 59 93 20)(25 61 79 45 29 65 83 37 33 69 75 41)(26 62 80 46 30 66 84 38 34 70 76 42)(27 63 81 47 31 67 73 39 35 71 77 43)(28 64 82 48 32 68 74 40 36 72 78 44)
(1 77 94 31)(2 78 95 32)(3 79 96 33)(4 80 85 34)(5 81 86 35)(6 82 87 36)(7 83 88 25)(8 84 89 26)(9 73 90 27)(10 74 91 28)(11 75 92 29)(12 76 93 30)(13 39 56 63)(14 40 57 64)(15 41 58 65)(16 42 59 66)(17 43 60 67)(18 44 49 68)(19 45 50 69)(20 46 51 70)(21 47 52 71)(22 48 53 72)(23 37 54 61)(24 38 55 62)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A ··· 6F 6G ··· 6O 12A ··· 12H 12I ··· 12AZ 12BA ··· 12BP order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 Q8 D6 C4○D4 C3×S3 Dic6 C4×S3 C3×Q8 S3×C6 C4○D12 C3×C4○D4 C3×Dic6 S3×C12 C3×C4○D12 kernel C12×Dic6 Dic3×C12 C3×Dic3⋊C4 C3×C4⋊Dic3 C122 C6×Dic6 C4×Dic6 C3×Dic6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C4×C12 C2×Dic6 Dic6 C4×C12 C3×C12 C2×C12 C3×C6 C42 C12 C12 C12 C2×C4 C6 C6 C4 C4 C2 # reps 1 2 2 1 1 1 2 8 4 4 2 2 2 16 1 2 3 2 2 4 4 4 6 4 4 8 8 8

Matrix representation of C12×Dic6 in GL3(𝔽13) generated by

 7 0 0 0 4 0 0 0 4
,
 1 0 0 0 2 0 0 0 7
,
 12 0 0 0 0 1 0 12 0
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] >;

C12×Dic6 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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