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## G = C6×Dic3⋊C4order 288 = 25·32

### Direct product of C6 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — C6×Dic3⋊C4
 Lower central C3 — C6 — C6×Dic3⋊C4
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C6×Dic3⋊C4
G = < a,b,c,d | a6=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 378 in 211 conjugacy classes, 114 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], C6 [×6], C6 [×8], C6 [×7], C2×C4 [×2], C2×C4 [×12], C23, C32, Dic3 [×4], Dic3 [×2], C12 [×14], C2×C6 [×2], C2×C6 [×12], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×4], C2×C12 [×20], C22×C6 [×2], C22×C6, C2×C4⋊C4, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], C62, C62 [×6], Dic3⋊C4 [×4], C3×C4⋊C4 [×4], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Dic3 [×8], C6×Dic3 [×2], C6×C12 [×2], C6×C12 [×2], C2×C62, C2×Dic3⋊C4, C6×C4⋊C4, C3×Dic3⋊C4 [×4], Dic3×C2×C6 [×2], C2×C6×C12, C6×Dic3⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3×S3, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C2×C4⋊C4, S3×C6 [×3], Dic3⋊C4 [×4], C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×Dic6 [×2], S3×C12 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C2×Dic3⋊C4, C6×C4⋊C4, C3×Dic3⋊C4 [×4], C6×Dic6, S3×C2×C12, C6×C3⋊D4, C6×Dic3⋊C4

Smallest permutation representation of C6×Dic3⋊C4
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 7 5 11 3 9)(2 8 6 12 4 10)(13 54 17 52 15 50)(14 49 18 53 16 51)(19 35 23 33 21 31)(20 36 24 34 22 32)(25 64 27 66 29 62)(26 65 28 61 30 63)(37 43 41 47 39 45)(38 44 42 48 40 46)(55 68 57 70 59 72)(56 69 58 71 60 67)(73 86 75 88 77 90)(74 87 76 89 78 85)(79 92 81 94 83 96)(80 93 82 95 84 91)
(1 58 11 67)(2 59 12 68)(3 60 7 69)(4 55 8 70)(5 56 9 71)(6 57 10 72)(13 92 52 83)(14 93 53 84)(15 94 54 79)(16 95 49 80)(17 96 50 81)(18 91 51 82)(19 74 33 89)(20 75 34 90)(21 76 35 85)(22 77 36 86)(23 78 31 87)(24 73 32 88)(25 43 66 39)(26 44 61 40)(27 45 62 41)(28 46 63 42)(29 47 64 37)(30 48 65 38)
(1 13 19 40)(2 14 20 41)(3 15 21 42)(4 16 22 37)(5 17 23 38)(6 18 24 39)(7 54 35 46)(8 49 36 47)(9 50 31 48)(10 51 32 43)(11 52 33 44)(12 53 34 45)(25 72 91 88)(26 67 92 89)(27 68 93 90)(28 69 94 85)(29 70 95 86)(30 71 96 87)(55 80 77 64)(56 81 78 65)(57 82 73 66)(58 83 74 61)(59 84 75 62)(60 79 76 63)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6N 6O ··· 6AI 12A ··· 12AF 12AG ··· 12AV order 1 2 ··· 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 2 2 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 Q8 D6 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 kernel C6×Dic3⋊C4 C3×Dic3⋊C4 Dic3×C2×C6 C2×C6×C12 C2×Dic3⋊C4 C6×Dic3 Dic3⋊C4 C22×Dic3 C22×C12 C2×Dic3 C22×C12 C62 C62 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 2 8 8 4 2 16 1 2 2 2 1 2 4 4 4 4 4 4 2 8 8 8

Matrix representation of C6×Dic3⋊C4 in GL4(𝔽13) generated by

 10 0 0 0 0 9 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 4 9 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 12 3 0 0 8 1
,
 1 0 0 0 0 5 0 0 0 0 8 2 0 0 0 5
G:=sub<GL(4,GF(13))| [10,0,0,0,0,9,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,9,10],[1,0,0,0,0,1,0,0,0,0,12,8,0,0,3,1],[1,0,0,0,0,5,0,0,0,0,8,0,0,0,2,5] >;

C6×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C6xDic3:C4");
// GroupNames label

G:=SmallGroup(288,694);
// by ID

G=gap.SmallGroup(288,694);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,1094,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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