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## G = S3×C4.Q8order 192 = 26·3

### Direct product of S3 and C4.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C4.Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — S3×C2×C8 — S3×C4.Q8
 Lower central C3 — C6 — C12 — S3×C4.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for S3×C4.Q8
G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=c-1d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 336 in 130 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.Q8, C4.Q8, C2×C4⋊C4, C22×C8, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×C4.Q8, C12.Q8, C8⋊Dic3, C3×C4.Q8, S3×C4⋊C4, S3×C2×C8, S3×C4.Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C4.Q8, C2×C4⋊C4, C2×SD16, S3×C2×C4, S3×D4, S3×Q8, C2×C4.Q8, S3×C4⋊C4, S3×SD16, S3×C4.Q8

Smallest permutation representation of S3×C4.Q8
On 96 points
Generators in S96
(1 82 17)(2 83 18)(3 84 19)(4 85 20)(5 86 21)(6 87 22)(7 88 23)(8 81 24)(9 34 63)(10 35 64)(11 36 57)(12 37 58)(13 38 59)(14 39 60)(15 40 61)(16 33 62)(25 89 47)(26 90 48)(27 91 41)(28 92 42)(29 93 43)(30 94 44)(31 95 45)(32 96 46)(49 70 77)(50 71 78)(51 72 79)(52 65 80)(53 66 73)(54 67 74)(55 68 75)(56 69 76)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(41 45)(42 46)(43 47)(44 48)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 65)(73 77)(74 78)(75 79)(76 80)
(1 47 5 43)(2 48 6 44)(3 41 7 45)(4 42 8 46)(9 75 13 79)(10 76 14 80)(11 77 15 73)(12 78 16 74)(17 89 21 93)(18 90 22 94)(19 91 23 95)(20 92 24 96)(25 86 29 82)(26 87 30 83)(27 88 31 84)(28 81 32 85)(33 54 37 50)(34 55 38 51)(35 56 39 52)(36 49 40 53)(57 70 61 66)(58 71 62 67)(59 72 63 68)(60 65 64 69)
(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 15 45 79)(2 10 46 74)(3 13 47 77)(4 16 48 80)(5 11 41 75)(6 14 42 78)(7 9 43 73)(8 12 44 76)(17 61 95 72)(18 64 96 67)(19 59 89 70)(20 62 90 65)(21 57 91 68)(22 60 92 71)(23 63 93 66)(24 58 94 69)(25 49 84 38)(26 52 85 33)(27 55 86 36)(28 50 87 39)(29 53 88 34)(30 56 81 37)(31 51 82 40)(32 54 83 35)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D4 D4 D6 D6 SD16 C4×S3 S3×Q8 S3×D4 S3×SD16 kernel S3×C4.Q8 C12.Q8 C8⋊Dic3 C3×C4.Q8 S3×C4⋊C4 S3×C2×C8 S3×C8 C4.Q8 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 D6 C8 C4 C22 C2 # reps 1 2 1 1 2 1 8 1 2 1 1 2 1 8 4 1 1 4

Matrix representation of S3×C4.Q8 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 71 0 0 0 0 1 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 24 0 0 0 0 6 1 0 0 0 0 0 0 0 61 0 0 0 0 6 61
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 2 0 0 0 0 36 0 0 0 0 0 0 0 0 32 0 0 0 0 16 0

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,6,0,0,0,0,24,1,0,0,0,0,0,0,0,6,0,0,0,0,61,61],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,36,0,0,0,0,2,0,0,0,0,0,0,0,0,16,0,0,0,0,32,0] >;

S3×C4.Q8 in GAP, Magma, Sage, TeX

S_3\times C_4.Q_8
% in TeX

G:=Group("S3xC4.Q8");
// GroupNames label

G:=SmallGroup(192,418);
// by ID

G=gap.SmallGroup(192,418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,555,58,438,102,6278]);
// Polycyclic

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

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