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## G = S3×Q8⋊C4order 192 = 26·3

### Direct product of S3 and Q8⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×Q8⋊C4
G = < a,b,c,d,e | a3=b2=c4=e4=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >

Subgroups: 424 in 162 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, S3×Q8, C6×Q8, C2×Q8⋊C4, C6.SD16, C2.Dic12, Q82Dic3, C3×Q8⋊C4, S3×C4⋊C4, S3×C2×C8, C2×S3×Q8, S3×Q8⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4×S3, C22×S3, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, S3×C2×C4, S3×D4, C2×Q8⋊C4, S3×C22⋊C4, S3×SD16, S3×Q16, S3×Q8⋊C4

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

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

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

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

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 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 D6 SD16 Q16 C4×S3 S3×D4 S3×D4 S3×SD16 S3×Q16 kernel S3×Q8⋊C4 C6.SD16 C2.Dic12 Q8⋊2Dic3 C3×Q8⋊C4 S3×C4⋊C4 S3×C2×C8 C2×S3×Q8 S3×Q8 Q8⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×Q8 D6 D6 Q8 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 4 4 1 1 2 2

Matrix representation of S3×Q8⋊C4 in GL4(𝔽73) generated by

 0 72 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 72 2 0 0 72 1
,
 72 0 0 0 0 72 0 0 0 0 60 24 0 0 72 13
,
 46 0 0 0 0 46 0 0 0 0 32 22 0 0 43 41
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,2,1],[72,0,0,0,0,72,0,0,0,0,60,72,0,0,24,13],[46,0,0,0,0,46,0,0,0,0,32,43,0,0,22,41] >;

S3×Q8⋊C4 in GAP, Magma, Sage, TeX

S_3\times Q_8\rtimes C_4
% in TeX

G:=Group("S3xQ8:C4");
// GroupNames label

G:=SmallGroup(192,360);
// by ID

G=gap.SmallGroup(192,360);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,58,851,438,102,6278]);
// Polycyclic

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

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