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## G = C6×C8⋊S3order 288 = 25·32

### Direct product of C6 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — S3×C2×C12 — C6×C8⋊S3
 Lower central C3 — C6 — C6×C8⋊S3
 Upper central C1 — C2×C12 — C2×C24

Generators and relations for C6×C8⋊S3
G = < a,b,c,d | a6=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 282 in 147 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8, C2×C8, M4(2) [×4], C22×C4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×4], C24 [×4], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C8⋊S3 [×4], C2×C3⋊C8, C2×C24 [×2], C2×C24 [×2], C3×M4(2) [×4], S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C3×C24 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, C2×C8⋊S3, C6×M4(2), C3×C8⋊S3 [×4], C6×C3⋊C8, C6×C24, S3×C2×C12, C6×C8⋊S3
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], M4(2) [×2], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), S3×C6 [×3], C8⋊S3 [×2], C3×M4(2) [×2], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, C2×C8⋊S3, C6×M4(2), C3×C8⋊S3 [×2], S3×C2×C12, C6×C8⋊S3

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

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I ··· 12T 12U 12V 12W 12X 24A ··· 24AF 24AG ··· 24AN order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 size 1 1 1 1 6 6 1 1 2 2 2 1 1 1 1 6 6 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 6 6 6 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 S3 D6 D6 M4(2) C3×S3 C4×S3 C4×S3 S3×C6 S3×C6 C8⋊S3 C3×M4(2) S3×C12 S3×C12 C3×C8⋊S3 kernel C6×C8⋊S3 C3×C8⋊S3 C6×C3⋊C8 C6×C24 S3×C2×C12 C2×C8⋊S3 S3×C12 C6×Dic3 S3×C2×C6 C8⋊S3 C2×C3⋊C8 C2×C24 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C2×C24 C24 C2×C12 C3×C6 C2×C8 C12 C2×C6 C8 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 1 1 2 4 2 2 8 2 2 2 8 4 4 1 2 1 4 2 2 2 4 2 8 8 4 4 16

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

 65 0 0 0 0 65 0 0 0 0 8 0 0 0 0 8
,
 46 0 0 0 0 46 0 0 0 0 63 17 0 0 0 10
,
 8 64 0 0 0 64 0 0 0 0 8 33 0 0 0 64
,
 72 0 0 0 63 1 0 0 0 0 28 71 0 0 63 45
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,8,0,0,0,0,8],[46,0,0,0,0,46,0,0,0,0,63,0,0,0,17,10],[8,0,0,0,64,64,0,0,0,0,8,0,0,0,33,64],[72,63,0,0,0,1,0,0,0,0,28,63,0,0,71,45] >;

C6×C8⋊S3 in GAP, Magma, Sage, TeX

C_6\times C_8\rtimes S_3
% in TeX

G:=Group("C6xC8:S3");
// GroupNames label

G:=SmallGroup(288,671);
// by ID

G=gap.SmallGroup(288,671);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,142,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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