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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, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C2×C4⋊C4, C3×Dic3, C3×C12, C62, C62, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C6×Dic3, C6×Dic3, C6×C12, C2×C62, C2×C4⋊Dic3, C6×C4⋊C4, C3×C4⋊Dic3, Dic3×C2×C6, C2×C6×C12, C6×C4⋊Dic3
Quotients:

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 29 19)(2 45 30 20)(3 46 25 21)(4 47 26 22)(5 48 27 23)(6 43 28 24)(7 17 38 55)(8 18 39 56)(9 13 40 57)(10 14 41 58)(11 15 42 59)(12 16 37 60)(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 28 11 26 9 30)(8 29 12 27 10 25)(13 20 17 24 15 22)(14 21 18 19 16 23)(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 77 26 49)(8 78 27 50)(9 73 28 51)(10 74 29 52)(11 75 30 53)(12 76 25 54)(13 72 24 82)(14 67 19 83)(15 68 20 84)(16 69 21 79)(17 70 22 80)(18 71 23 81)(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), (1,44,29,19)(2,45,30,20)(3,46,25,21)(4,47,26,22)(5,48,27,23)(6,43,28,24)(7,17,38,55)(8,18,39,56)(9,13,40,57)(10,14,41,58)(11,15,42,59)(12,16,37,60)(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)(7,28,11,26,9,30)(8,29,12,27,10,25)(13,20,17,24,15,22)(14,21,18,19,16,23)(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)(7,77,26,49)(8,78,27,50)(9,73,28,51)(10,74,29,52)(11,75,30,53)(12,76,25,54)(13,72,24,82)(14,67,19,83)(15,68,20,84)(16,69,21,79)(17,70,22,80)(18,71,23,81)(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), (1,44,29,19)(2,45,30,20)(3,46,25,21)(4,47,26,22)(5,48,27,23)(6,43,28,24)(7,17,38,55)(8,18,39,56)(9,13,40,57)(10,14,41,58)(11,15,42,59)(12,16,37,60)(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)(7,28,11,26,9,30)(8,29,12,27,10,25)(13,20,17,24,15,22)(14,21,18,19,16,23)(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)(7,77,26,49)(8,78,27,50)(9,73,28,51)(10,74,29,52)(11,75,30,53)(12,76,25,54)(13,72,24,82)(14,67,19,83)(15,68,20,84)(16,69,21,79)(17,70,22,80)(18,71,23,81)(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)], [(1,44,29,19),(2,45,30,20),(3,46,25,21),(4,47,26,22),(5,48,27,23),(6,43,28,24),(7,17,38,55),(8,18,39,56),(9,13,40,57),(10,14,41,58),(11,15,42,59),(12,16,37,60),(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),(7,28,11,26,9,30),(8,29,12,27,10,25),(13,20,17,24,15,22),(14,21,18,19,16,23),(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),(7,77,26,49),(8,78,27,50),(9,73,28,51),(10,74,29,52),(11,75,30,53),(12,76,25,54),(13,72,24,82),(14,67,19,83),(15,68,20,84),(16,69,21,79),(17,70,22,80),(18,71,23,81),(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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