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## G = C6×C4⋊Dic3order 288 = 25·32

### Direct product of C6 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C4⋊Dic3
G = < a,b,c,d | a6=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 378 in 211 conjugacy classes, 130 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×6], C6 [×6], C6 [×8], C6 [×7], C2×C4 [×6], C2×C4 [×8], C23, C32, Dic3 [×4], C12 [×8], C12 [×8], 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 [×4], C2×Dic3 [×4], C2×C12 [×12], C2×C12 [×14], C22×C6 [×2], C22×C6, C2×C4⋊C4, C3×Dic3 [×4], C3×C12 [×4], C62, C62 [×6], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C6×C12 [×6], C2×C62, C2×C4⋊Dic3, C6×C4⋊C4, C3×C4⋊Dic3 [×4], Dic3×C2×C6 [×2], C2×C6×C12, C6×C4⋊Dic3
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3×S3, Dic6 [×2], D12 [×2], C2×Dic3 [×6], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C2×C4⋊C4, C3×Dic3 [×4], S3×C6 [×3], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C2×Dic6, C2×D12, C22×Dic3, C22×C12, C6×D4, C6×Q8, C3×Dic6 [×2], C3×D12 [×2], C6×Dic3 [×6], S3×C2×C6, C2×C4⋊Dic3, C6×C4⋊C4, C3×C4⋊Dic3 [×4], C6×Dic6, C6×D12, Dic3×C2×C6, C6×C4⋊Dic3

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

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

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

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

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 Dic3 D6 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 S3×C6 C3×Dic6 C3×D12 kernel C6×C4⋊Dic3 C3×C4⋊Dic3 Dic3×C2×C6 C2×C6×C12 C2×C4⋊Dic3 C6×C12 C4⋊Dic3 C22×Dic3 C22×C12 C2×C12 C22×C12 C62 C62 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 2 8 8 4 2 16 1 2 2 4 2 1 2 4 4 4 4 8 4 2 8 8

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

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

C6×C4⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C6xC4:Dic3");
// GroupNames label

G:=SmallGroup(288,696);
// by ID

G=gap.SmallGroup(288,696);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,1094,268,9414]);
// Polycyclic

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

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