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## G = S3×C4⋊Q8order 192 = 26·3

### Direct product of S3 and C4⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C4⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C42 — S3×C4⋊Q8
 Lower central C3 — C2×C6 — S3×C4⋊Q8
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for S3×C4⋊Q8
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=c-1, ede-1=d-1 >

Subgroups: 656 in 290 conjugacy classes, 131 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×14], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×27], Q8 [×16], C23, Dic3 [×6], Dic3 [×4], C12 [×6], C12 [×4], D6 [×6], C2×C6, C42, C42 [×3], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×7], C2×Q8 [×2], C2×Q8 [×14], Dic6 [×12], C4×S3 [×12], C4×S3 [×8], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C2×C42, C2×C4⋊C4 [×4], C4⋊Q8, C4⋊Q8 [×7], C22×Q8 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C4×C12, C3×C4⋊C4 [×4], C2×Dic6 [×6], S3×C2×C4, S3×C2×C4 [×6], S3×Q8 [×8], C6×Q8 [×2], C2×C4⋊Q8, C122Q8, S3×C42, C12⋊Q8 [×4], S3×C4⋊C4 [×4], Dic3⋊Q8 [×2], C3×C4⋊Q8, C2×S3×Q8 [×2], S3×C4⋊Q8
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×8], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×12], C24, C22×S3 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], S3×D4 [×2], S3×Q8 [×4], S3×C23, C2×C4⋊Q8, C2×S3×D4, C2×S3×Q8 [×2], S3×C4⋊Q8

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

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

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

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

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

Matrix representation of S3×C4⋊Q8 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 4 0 0 0 0 4 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 4 0 0 0 0 4 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 9 0 0 0 0 9 3
,
 4 3 0 0 0 0 3 9 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 12 0

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

S3×C4⋊Q8 in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes Q_8
% in TeX

G:=Group("S3xC4:Q8");
// GroupNames label

G:=SmallGroup(192,1282);
// by ID

G=gap.SmallGroup(192,1282);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,570,185,80,6278]);
// 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^-1,e*d*e^-1=d^-1>;
// generators/relations

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