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## G = S3×C22.D4order 192 = 26·3

### Direct product of S3 and C22.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C22.D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — S3×C22.D4
 Lower central C3 — C2×C6 — S3×C22.D4
 Upper central C1 — C22 — C22.D4

Generators and relations for S3×C22.D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=de-1 >

Subgroups: 944 in 342 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C22.D4, C23×C4, C22×D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C2×C22.D4, S3×C22⋊C4, S3×C22⋊C4, C23.9D6, C23.21D6, S3×C4⋊C4, D6.D4, C23.28D6, C23.23D6, C3×C22.D4, S3×C22×C4, C2×S3×D4, S3×C22.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22.D4, C22×D4, C2×C4○D4, S3×D4, S3×C23, C2×C22.D4, C2×S3×D4, S3×C4○D4, S3×C22.D4

Smallest permutation representation of S3×C22.D4
On 48 points
Generators in S48
(1 43 22)(2 44 23)(3 41 24)(4 42 21)(5 29 36)(6 30 33)(7 31 34)(8 32 35)(9 39 13)(10 40 14)(11 37 15)(12 38 16)(17 27 48)(18 28 45)(19 25 46)(20 26 47)
(1 3)(2 4)(5 7)(6 8)(9 37)(10 38)(11 39)(12 40)(13 15)(14 16)(17 46)(18 47)(19 48)(20 45)(21 44)(22 41)(23 42)(24 43)(25 27)(26 28)(29 34)(30 35)(31 36)(32 33)
(1 15)(2 25)(3 13)(4 27)(5 14)(6 28)(7 16)(8 26)(9 41)(10 29)(11 43)(12 31)(17 21)(18 33)(19 23)(20 35)(22 37)(24 39)(30 45)(32 47)(34 38)(36 40)(42 48)(44 46)
(1 6)(2 7)(3 8)(4 5)(9 47)(10 48)(11 45)(12 46)(13 26)(14 27)(15 28)(16 25)(17 40)(18 37)(19 38)(20 39)(21 36)(22 33)(23 34)(24 35)(29 42)(30 43)(31 44)(32 41)
(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)
(2 5)(4 7)(9 47)(10 12)(11 45)(13 26)(14 16)(15 28)(17 19)(18 37)(20 39)(21 34)(23 36)(25 27)(29 44)(31 42)(38 40)(46 48)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 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 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 2 2 3 3 3 3 4 6 6 12 2 2 2 2 2 4 4 4 6 6 6 6 12 12 12 2 2 2 4 4 8 4 4 4 4 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 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 S3×D4 S3×C4○D4 kernel S3×C22.D4 S3×C22⋊C4 C23.9D6 C23.21D6 S3×C4⋊C4 D6.D4 C23.28D6 C23.23D6 C3×C22.D4 S3×C22×C4 C2×S3×D4 C22.D4 C22×S3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C22 C2 # reps 1 3 2 1 2 2 1 1 1 1 1 1 4 3 2 1 1 8 2 4

Matrix representation of S3×C22.D4 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 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 5 8 0 0 0 0 10 8 0 0 0 0 0 0 0 8 0 0 0 0 5 0 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 10 8 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 2 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

S3×C22.D4 in GAP, Magma, Sage, TeX

S_3\times C_2^2.D_4
% in TeX

G:=Group("S3xC2^2.D4");
// GroupNames label

G:=SmallGroup(192,1211);
// by ID

G=gap.SmallGroup(192,1211);
# by ID

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

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

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