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## G = S3×C42.C2order 192 = 26·3

### Direct product of S3 and C42.C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C42.C2
G = < a,b,c,d,e | a3=b2=c4=d4=1, e2=d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd2, ede-1=c2d >

Subgroups: 480 in 226 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C42, C2×C4⋊C4, C42.C2, C42.C2, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×C42.C2, C12.6Q8, S3×C42, Dic3.Q8, C4.Dic6, S3×C4⋊C4, C3×C42.C2, S3×C42.C2
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C42.C2, C22×Q8, C2×C4○D4, S3×Q8, S3×C23, C2×C42.C2, C2×S3×Q8, S3×C4○D4, S3×C42.C2

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 4Q 4R 4S 4T 6A 6B 6C 12A ··· 12F 12G 12H 12I 12J 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 12 ··· 12 12 12 12 12 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 4 ··· 4 8 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 C4○D4 S3×Q8 S3×C4○D4 kernel S3×C42.C2 C12.6Q8 S3×C42 Dic3.Q8 C4.Dic6 S3×C4⋊C4 C3×C42.C2 C42.C2 C4×S3 C42 C4⋊C4 D6 C4 C2 # reps 1 1 1 4 2 6 1 1 4 1 6 8 2 4

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 11 4 0 0 0 0 9 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 10 0 0 0 0 10 4
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 4 3 0 0 0 0 3 9
,
 9 2 0 0 0 0 11 4 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 9 0 0 0 0 9 10

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,10,0,0,0,0,10,4],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,3,0,0,0,0,3,9],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,9,0,0,0,0,9,10] >;

S3×C42.C2 in GAP, Magma, Sage, TeX

S_3\times C_4^2.C_2
% in TeX

G:=Group("S3xC4^2.C2");
// GroupNames label

G:=SmallGroup(192,1246);
// by ID

G=gap.SmallGroup(192,1246);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,346,297,136,6278]);
// Polycyclic

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

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