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## G = C3×C8.6D6order 288 = 25·32

### Direct product of C3 and C8.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C8.6D6
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×D24 — C3×C8.6D6
 Lower central C3 — C6 — C12 — C24 — C3×C8.6D6
 Upper central C1 — C6 — C12 — C24 — C3×Q16

Generators and relations for C3×C8.6D6
G = < a,b,c,d | a3=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 202 in 61 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C32, C12 [×2], C12 [×5], D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24 [×2], C24, D12, C3×D4, C3×Q8 [×4], SD32, C3×C12, C3×C12, S3×C6, C3⋊C16, C48, D24, C3×D8, C3×Q16 [×2], C3×Q16, C3×C24, C3×D12, Q8×C32, C8.6D6, C3×SD32, C3×C3⋊C16, C3×D24, C32×Q16, C3×C8.6D6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C8.6D6, C3×SD32, C3×D4⋊S3, C3×C8.6D6

Smallest permutation representation of C3×C8.6D6
On 96 points
Generators in S96
(1 63 36)(2 64 37)(3 49 38)(4 50 39)(5 51 40)(6 52 41)(7 53 42)(8 54 43)(9 55 44)(10 56 45)(11 57 46)(12 58 47)(13 59 48)(14 60 33)(15 61 34)(16 62 35)(17 72 91)(18 73 92)(19 74 93)(20 75 94)(21 76 95)(22 77 96)(23 78 81)(24 79 82)(25 80 83)(26 65 84)(27 66 85)(28 67 86)(29 68 87)(30 69 88)(31 70 89)(32 71 90)
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)(49 55 61 51 57 63 53 59)(50 56 62 52 58 64 54 60)(65 71 77 67 73 79 69 75)(66 72 78 68 74 80 70 76)(81 87 93 83 89 95 85 91)(82 88 94 84 90 96 86 92)
(1 75 55 86 36 20 9 67 63 94 44 28)(2 27 45 93 64 66 10 19 37 85 56 74)(3 73 57 84 38 18 11 65 49 92 46 26)(4 25 47 91 50 80 12 17 39 83 58 72)(5 71 59 82 40 32 13 79 51 90 48 24)(6 23 33 89 52 78 14 31 41 81 60 70)(7 69 61 96 42 30 15 77 53 88 34 22)(8 21 35 87 54 76 16 29 43 95 62 68)
(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)

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

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

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

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 12E 12F ··· 12M 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 8 8 12 12 12 12 12 12 ··· 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 24 1 1 2 2 2 2 8 1 1 2 2 2 24 24 2 2 2 2 4 4 4 8 ··· 8 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 SD32 S3×C6 C3×D8 C3×C3⋊D4 C3×SD32 D4⋊S3 C8.6D6 C3×D4⋊S3 C3×C8.6D6 kernel C3×C8.6D6 C3×C3⋊C16 C3×D24 C32×Q16 C8.6D6 C3⋊C16 D24 C3×Q16 C3×Q16 C3×C12 C24 C3×C6 Q16 C12 C12 C32 C8 C6 C4 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 8 1 2 2 4

Matrix representation of C3×C8.6D6 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 6 3 0 2 1 6 1 1 2 2 2 3 3 4 2 6
,
 2 6 5 4 5 2 3 5 6 2 6 3 1 3 4 4
,
 2 0 6 6 5 6 0 6 4 6 4 1 4 5 1 2
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,5,6,1,6,2,2,3,5,3,6,4,4,5,3,4],[2,5,4,4,0,6,6,5,6,0,4,1,6,6,1,2] >;

C3×C8.6D6 in GAP, Magma, Sage, TeX

C_3\times C_8._6D_6
% in TeX

G:=Group("C3xC8.6D6");
// GroupNames label

G:=SmallGroup(288,262);
// by ID

G=gap.SmallGroup(288,262);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,344,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^4,d^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^-1*c^5>;
// generators/relations

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