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## G = C6.(S3×C8)  order 288 = 25·32

### 3rd non-split extension by C6 of S3×C8 acting via S3×C8/C3⋊C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6.(S3×C8)
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×C3⋊C8 — C6.(S3×C8)
 Lower central C32 — C6.(S3×C8)
 Upper central C1 — C2×C4

Generators and relations for C6.(S3×C8)
G = < a,b,c,d | a6=b8=c3=1, d2=a3, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 266 in 103 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×12], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C4×C8, C3⋊Dic3 [×4], C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×Dic3 [×3], C2×C24 [×2], C3×C3⋊C8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C8×Dic3 [×2], C6×C3⋊C8 [×2], C4×C3⋊Dic3, C6.(S3×C8)
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×2], C8 [×4], C2×C4 [×3], Dic3 [×4], D6 [×2], C42, C2×C8 [×2], C4×S3 [×4], C2×Dic3 [×2], C4×C8, S32, S3×C8 [×4], C4×Dic3 [×2], S3×Dic3 [×2], C6.D6, C8×Dic3 [×2], C12.29D6 [×2], Dic32, C6.(S3×C8)

Smallest permutation representation of C6.(S3×C8)
On 96 points
Generators in S96
(1 34 67 22 32 95)(2 96 25 23 68 35)(3 36 69 24 26 89)(4 90 27 17 70 37)(5 38 71 18 28 91)(6 92 29 19 72 39)(7 40 65 20 30 93)(8 94 31 21 66 33)(9 47 83 77 63 56)(10 49 64 78 84 48)(11 41 85 79 57 50)(12 51 58 80 86 42)(13 43 87 73 59 52)(14 53 60 74 88 44)(15 45 81 75 61 54)(16 55 62 76 82 46)
(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 32 67)(2 25 68)(3 26 69)(4 27 70)(5 28 71)(6 29 72)(7 30 65)(8 31 66)(9 83 63)(10 84 64)(11 85 57)(12 86 58)(13 87 59)(14 88 60)(15 81 61)(16 82 62)(17 37 90)(18 38 91)(19 39 92)(20 40 93)(21 33 94)(22 34 95)(23 35 96)(24 36 89)(41 79 50)(42 80 51)(43 73 52)(44 74 53)(45 75 54)(46 76 55)(47 77 56)(48 78 49)
(1 53 22 88)(2 54 23 81)(3 55 24 82)(4 56 17 83)(5 49 18 84)(6 50 19 85)(7 51 20 86)(8 52 21 87)(9 27 77 37)(10 28 78 38)(11 29 79 39)(12 30 80 40)(13 31 73 33)(14 32 74 34)(15 25 75 35)(16 26 76 36)(41 92 57 72)(42 93 58 65)(43 94 59 66)(44 95 60 67)(45 96 61 68)(46 89 62 69)(47 90 63 70)(48 91 64 71)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G 6H 6I 8A ··· 8P 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 9 ··· 9 2 ··· 2 4 4 4 3 ··· 3 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + - + + - + image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 C4×S3 C4×S3 S3×C8 S32 S3×Dic3 C6.D6 C12.29D6 kernel C6.(S3×C8) C6×C3⋊C8 C4×C3⋊Dic3 C3×C3⋊C8 C2×C3⋊Dic3 C3⋊Dic3 C2×C3⋊C8 C3⋊C8 C2×C12 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 16 2 4 2 4 4 16 1 2 1 4

Matrix representation of C6.(S3×C8) in GL4(𝔽73) generated by

 1 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 27 46 0 0 0 46 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 46 27 0 0 0 27 0 0 0 0 72 1 0 0 0 1
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,46,46,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[46,0,0,0,27,27,0,0,0,0,72,0,0,0,1,1] >;

C6.(S3×C8) in GAP, Magma, Sage, TeX

C_6.(S_3\times C_8)
% in TeX

G:=Group("C6.(S3xC8)");
// GroupNames label

G:=SmallGroup(288,201);
// by ID

G=gap.SmallGroup(288,201);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,92,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^8=c^3=1,d^2=a^3,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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