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## G = S3×C22⋊Q8order 192 = 26·3

### Direct product of S3 and C22⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C22⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — S3×C22⋊Q8
 Lower central C3 — C2×C6 — S3×C22⋊Q8
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for S3×C22⋊Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 800 in 322 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C23×C4, C22×Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×C2×C4, S3×Q8, C22×Dic3, C22×C12, C6×Q8, S3×C23, C2×C22⋊Q8, Dic3.D4, S3×C22⋊C4, S3×C4⋊C4, S3×C4⋊C4, D6⋊Q8, C4.D12, C12.48D4, D63Q8, C3×C22⋊Q8, S3×C22×C4, C2×S3×Q8, S3×C22⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, S3×D4, S3×Q8, S3×C23, C2×C22⋊Q8, C2×S3×D4, C2×S3×Q8, S3×C4○D4, S3×C22⋊Q8

Smallest permutation representation of S3×C22⋊Q8
On 48 points
Generators in S48
(1 32 47)(2 29 48)(3 30 45)(4 31 46)(5 44 14)(6 41 15)(7 42 16)(8 43 13)(9 19 39)(10 20 40)(11 17 37)(12 18 38)(21 33 25)(22 34 26)(23 35 27)(24 36 28)
(1 9)(2 10)(3 11)(4 12)(5 34)(6 35)(7 36)(8 33)(13 25)(14 26)(15 27)(16 28)(17 45)(18 46)(19 47)(20 48)(21 43)(22 44)(23 41)(24 42)(29 40)(30 37)(31 38)(32 39)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 32)(18 29)(19 30)(20 31)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)(37 47)(38 48)(39 45)(40 46)(41 43)(42 44)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)(33 43)(34 44)(35 41)(36 42)(37 45)(38 46)(39 47)(40 48)
(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)
(1 26 3 28)(2 25 4 27)(5 17 7 19)(6 20 8 18)(9 14 11 16)(10 13 12 15)(21 31 23 29)(22 30 24 32)(33 46 35 48)(34 45 36 47)(37 42 39 44)(38 41 40 43)

G:=sub<Sym(48)| (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,14)(6,41,15)(7,42,16)(8,43,13)(9,19,39)(10,20,40)(11,17,37)(12,18,38)(21,33,25)(22,34,26)(23,35,27)(24,36,28), (1,9)(2,10)(3,11)(4,12)(5,34)(6,35)(7,36)(8,33)(13,25)(14,26)(15,27)(16,28)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,40)(30,37)(31,38)(32,39), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36)(37,47)(38,48)(39,45)(40,46)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (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), (1,26,3,28)(2,25,4,27)(5,17,7,19)(6,20,8,18)(9,14,11,16)(10,13,12,15)(21,31,23,29)(22,30,24,32)(33,46,35,48)(34,45,36,47)(37,42,39,44)(38,41,40,43)>;

G:=Group( (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,14)(6,41,15)(7,42,16)(8,43,13)(9,19,39)(10,20,40)(11,17,37)(12,18,38)(21,33,25)(22,34,26)(23,35,27)(24,36,28), (1,9)(2,10)(3,11)(4,12)(5,34)(6,35)(7,36)(8,33)(13,25)(14,26)(15,27)(16,28)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,40)(30,37)(31,38)(32,39), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36)(37,47)(38,48)(39,45)(40,46)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (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), (1,26,3,28)(2,25,4,27)(5,17,7,19)(6,20,8,18)(9,14,11,16)(10,13,12,15)(21,31,23,29)(22,30,24,32)(33,46,35,48)(34,45,36,47)(37,42,39,44)(38,41,40,43) );

G=PermutationGroup([[(1,32,47),(2,29,48),(3,30,45),(4,31,46),(5,44,14),(6,41,15),(7,42,16),(8,43,13),(9,19,39),(10,20,40),(11,17,37),(12,18,38),(21,33,25),(22,34,26),(23,35,27),(24,36,28)], [(1,9),(2,10),(3,11),(4,12),(5,34),(6,35),(7,36),(8,33),(13,25),(14,26),(15,27),(16,28),(17,45),(18,46),(19,47),(20,48),(21,43),(22,44),(23,41),(24,42),(29,40),(30,37),(31,38),(32,39)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,32),(18,29),(19,30),(20,31),(21,23),(22,24),(25,27),(26,28),(33,35),(34,36),(37,47),(38,48),(39,45),(40,46),(41,43),(42,44)], [(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29),(33,43),(34,44),(35,41),(36,42),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,26,3,28),(2,25,4,27),(5,17,7,19),(6,20,8,18),(9,14,11,16),(10,13,12,15),(21,31,23,29),(22,30,24,32),(33,46,35,48),(34,45,36,47),(37,42,39,44),(38,41,40,43)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 1 1 2 2 3 3 3 3 6 6 2 2 2 2 2 4 4 4 4 6 6 6 6 12 12 12 12 2 2 2 4 4 4 4 4 4 8 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 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 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 D6 C4○D4 S3×D4 S3×Q8 S3×C4○D4 kernel S3×C22⋊Q8 Dic3.D4 S3×C22⋊C4 S3×C4⋊C4 D6⋊Q8 C4.D12 C12.48D4 D6⋊3Q8 C3×C22⋊Q8 S3×C22×C4 C2×S3×Q8 C22⋊Q8 C4×S3 C22×S3 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 D6 C4 C22 C2 # reps 1 2 2 3 2 1 1 1 1 1 1 1 4 4 2 3 1 1 4 2 2 2

Matrix representation of S3×C22⋊Q8 in GL6(𝔽13)

 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 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 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 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 2 12 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 5 0 0 0 0 3 6 0 0 0 0 0 0 5 0 0 0 0 0 10 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 6 0 0 0 0 4 12 0 0 0 0 0 0 5 8 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,3,0,0,0,0,5,6,0,0,0,0,0,0,5,10,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,4,0,0,0,0,6,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S3×C22⋊Q8 in GAP, Magma, Sage, TeX

S_3\times C_2^2\rtimes Q_8
% in TeX

G:=Group("S3xC2^2:Q8");
// GroupNames label

G:=SmallGroup(192,1185);
// by ID

G=gap.SmallGroup(192,1185);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,794,297,136,6278]);
// Polycyclic

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

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