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## G = S3×C2.C42order 192 = 26·3

### Direct product of S3 and C2.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C2.C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C22×C4 — S3×C2.C42
 Lower central C3 — C6 — S3×C2.C42
 Upper central C1 — C23 — C2.C42

Generators and relations for S3×C2.C42
G = < a,b,c,d,e | a3=b2=c2=d4=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >

Subgroups: 800 in 330 conjugacy classes, 131 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2.C42, C2.C42, C23×C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C2.C42, C6.C42, C3×C2.C42, S3×C22×C4, S3×C2.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, C2×C2.C42, S3×C42, S3×C22⋊C4, S3×C4⋊C4, S3×C2.C42

Smallest permutation representation of S3×C2.C42
On 96 points
Generators in S96
(1 35 9)(2 36 10)(3 33 11)(4 34 12)(5 60 55)(6 57 56)(7 58 53)(8 59 54)(13 67 39)(14 68 40)(15 65 37)(16 66 38)(17 89 87)(18 90 88)(19 91 85)(20 92 86)(21 71 47)(22 72 48)(23 69 45)(24 70 46)(25 79 50)(26 80 51)(27 77 52)(28 78 49)(29 63 81)(30 64 82)(31 61 83)(32 62 84)(41 75 93)(42 76 94)(43 73 95)(44 74 96)
(1 29)(2 30)(3 31)(4 32)(5 80)(6 77)(7 78)(8 79)(9 63)(10 64)(11 61)(12 62)(13 17)(14 18)(15 19)(16 20)(21 75)(22 76)(23 73)(24 74)(25 59)(26 60)(27 57)(28 58)(33 83)(34 84)(35 81)(36 82)(37 91)(38 92)(39 89)(40 90)(41 71)(42 72)(43 69)(44 70)(45 95)(46 96)(47 93)(48 94)(49 53)(50 54)(51 55)(52 56)(65 85)(66 86)(67 87)(68 88)
(1 31)(2 32)(3 29)(4 30)(5 78)(6 79)(7 80)(8 77)(9 83)(10 84)(11 81)(12 82)(13 91)(14 92)(15 89)(16 90)(17 37)(18 38)(19 39)(20 40)(21 95)(22 96)(23 93)(24 94)(25 56)(26 53)(27 54)(28 55)(33 63)(34 64)(35 61)(36 62)(41 69)(42 70)(43 71)(44 72)(45 75)(46 76)(47 73)(48 74)(49 60)(50 57)(51 58)(52 59)(65 87)(66 88)(67 85)(68 86)
(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 77 65 71)(2 5 66 44)(3 79 67 69)(4 7 68 42)(6 85 41 29)(8 87 43 31)(9 27 15 21)(10 55 16 96)(11 25 13 23)(12 53 14 94)(17 73 61 59)(18 48 62 49)(19 75 63 57)(20 46 64 51)(22 84 28 90)(24 82 26 92)(30 80 86 70)(32 78 88 72)(33 50 39 45)(34 58 40 76)(35 52 37 47)(36 60 38 74)(54 89 95 83)(56 91 93 81)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4L 4M ··· 4X 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 S3 D4 Q8 D6 C4×S3 S3×D4 S3×Q8 kernel S3×C2.C42 C6.C42 C3×C2.C42 S3×C22×C4 S3×C2×C4 C2.C42 C22×S3 C22×S3 C22×C4 C2×C4 C22 C22 # reps 1 3 1 3 24 1 6 2 3 12 3 1

Matrix representation of S3×C2.C42 in GL6(𝔽13)

 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;

S3×C2.C42 in GAP, Magma, Sage, TeX

S_3\times C_2.C_4^2
% in TeX

G:=Group("S3xC2.C4^2");
// GroupNames label

G:=SmallGroup(192,222);
// by ID

G=gap.SmallGroup(192,222);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,387,58,6278]);
// Polycyclic

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

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