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## G = S3×C2.D8order 192 = 26·3

### Direct product of S3 and C2.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C2.D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — S3×C2×C8 — S3×C2.D8
 Lower central C3 — C6 — C12 — S3×C2.D8
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for S3×C2.D8
G = < a,b,c,d,e | a3=b2=c2=d8=1, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 336 in 130 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, 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, C2.D8, C2.D8, 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×C2.D8, C6.Q16, C241C4, C3×C2.D8, S3×C4⋊C4, S3×C2×C8, S3×C2.D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, S3×C2×C4, S3×D4, S3×Q8, C2×C2.D8, S3×C4⋊C4, S3×D8, S3×Q16, S3×C2.D8

Smallest permutation representation of S3×C2.D8
On 96 points
Generators in S96
(1 34 79)(2 35 80)(3 36 73)(4 37 74)(5 38 75)(6 39 76)(7 40 77)(8 33 78)(9 24 84)(10 17 85)(11 18 86)(12 19 87)(13 20 88)(14 21 81)(15 22 82)(16 23 83)(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 61)(50 71 62)(51 72 63)(52 65 64)(53 66 57)(54 67 58)(55 68 59)(56 69 60)
(1 5)(2 6)(3 7)(4 8)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 70)(58 71)(59 72)(60 65)(61 66)(62 67)(63 68)(64 69)(81 85)(82 86)(83 87)(84 88)
(1 45)(2 46)(3 47)(4 48)(5 41)(6 42)(7 43)(8 44)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 84)(50 85)(51 86)(52 87)(53 88)(54 81)(55 82)(56 83)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 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 56 45 83)(2 55 46 82)(3 54 47 81)(4 53 48 88)(5 52 41 87)(6 51 42 86)(7 50 43 85)(8 49 44 84)(9 33 70 30)(10 40 71 29)(11 39 72 28)(12 38 65 27)(13 37 66 26)(14 36 67 25)(15 35 68 32)(16 34 69 31)(17 77 62 93)(18 76 63 92)(19 75 64 91)(20 74 57 90)(21 73 58 89)(22 80 59 96)(23 79 60 95)(24 78 61 94)

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

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

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

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 2 4 4 4 4 type + + + + + + + - + + + + + - - + + - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D4 D4 D6 D6 D8 Q16 C4×S3 S3×Q8 S3×D4 S3×D8 S3×Q16 kernel S3×C2.D8 C6.Q16 C24⋊1C4 C3×C2.D8 S3×C4⋊C4 S3×C2×C8 S3×C8 C2.D8 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 D6 D6 C8 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 2 1 1 2 1 4 4 4 1 1 2 2

Matrix representation of S3×C2.D8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 1 0
,
 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 72 72
,
 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 32 38 0 0 0 48 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 46 0 0 0 0 0 1 67 0 0 0 49 72 0 0 0 0 0 72 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,1,0,0,0,72,0],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,72],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,32,48,0,0,0,38,0,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,1,49,0,0,0,67,72,0,0,0,0,0,72,0,0,0,0,0,72] >;

S3×C2.D8 in GAP, Magma, Sage, TeX

S_3\times C_2.D_8
% in TeX

G:=Group("S3xC2.D8");
// GroupNames label

G:=SmallGroup(192,438);
// by ID

G=gap.SmallGroup(192,438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,58,438,102,6278]);
// Polycyclic

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

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