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## G = S3×Q32order 192 = 26·3

### Direct product of S3 and Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — S3×Q32
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×Q16 — S3×Q32
 Lower central C3 — C6 — C12 — C24 — S3×Q32
 Upper central C1 — C2 — C4 — C8 — Q32

Generators and relations for S3×Q32
G = < a,b,c,d | a3=b2=c16=1, d2=c8, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 252 in 82 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, Q8, Dic3, Dic3, C12, C12, D6, C16, C16, C2×C8, Q16, Q16, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, C3×Q8, C2×C16, Q32, Q32, C2×Q16, C3⋊C16, C48, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, C2×Q32, S3×C16, Dic24, C3⋊Q32, C3×Q32, S3×Q16, S3×Q32
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, Q32, C2×D8, S3×D4, C2×Q32, S3×D8, S3×Q32

Smallest permutation representation of S3×Q32
On 96 points
Generators in S96
(1 46 72)(2 47 73)(3 48 74)(4 33 75)(5 34 76)(6 35 77)(7 36 78)(8 37 79)(9 38 80)(10 39 65)(11 40 66)(12 41 67)(13 42 68)(14 43 69)(15 44 70)(16 45 71)(17 51 89)(18 52 90)(19 53 91)(20 54 92)(21 55 93)(22 56 94)(23 57 95)(24 58 96)(25 59 81)(26 60 82)(27 61 83)(28 62 84)(29 63 85)(30 64 86)(31 49 87)(32 50 88)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 65)(48 66)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 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 89 9 81)(2 88 10 96)(3 87 11 95)(4 86 12 94)(5 85 13 93)(6 84 14 92)(7 83 15 91)(8 82 16 90)(17 38 25 46)(18 37 26 45)(19 36 27 44)(20 35 28 43)(21 34 29 42)(22 33 30 41)(23 48 31 40)(24 47 32 39)(49 66 57 74)(50 65 58 73)(51 80 59 72)(52 79 60 71)(53 78 61 70)(54 77 62 69)(55 76 63 68)(56 75 64 67)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 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 S3 D4 D4 D6 D6 D8 D8 Q32 S3×D4 S3×D8 S3×Q32 kernel S3×Q32 S3×C16 Dic24 C3⋊Q32 C3×Q32 S3×Q16 Q32 C3⋊C8 C4×S3 C16 Q16 Dic3 D6 S3 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 1 1 2 2 2 8 1 2 4

Matrix representation of S3×Q32 in GL4(𝔽97) generated by

 96 96 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 96 96 0 0 0 0 96 0 0 0 0 96
,
 1 0 0 0 0 1 0 0 0 0 24 91 0 0 66 28
,
 1 0 0 0 0 1 0 0 0 0 46 75 0 0 83 51
G:=sub<GL(4,GF(97))| [96,1,0,0,96,0,0,0,0,0,1,0,0,0,0,1],[1,96,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[1,0,0,0,0,1,0,0,0,0,24,66,0,0,91,28],[1,0,0,0,0,1,0,0,0,0,46,83,0,0,75,51] >;

S3×Q32 in GAP, Magma, Sage, TeX

S_3\times Q_{32}
% in TeX

G:=Group("S3xQ32");
// GroupNames label

G:=SmallGroup(192,476);
// by ID

G=gap.SmallGroup(192,476);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,135,184,346,185,192,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^16=1,d^2=c^8,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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