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## G = S3×C22≀C2order 192 = 26·3

### Direct product of S3 and C22≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C22≀C2
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — S3×C22≀C2
 Lower central C3 — C2×C6 — S3×C22≀C2
 Upper central C1 — C22 — C22≀C2

Generators and relations for S3×C22≀C2
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 2064 in 662 conjugacy classes, 131 normal (16 characteristic)
C1, C2 [×3], C2 [×18], C3, C4 [×6], C22, C22 [×6], C22 [×90], S3 [×4], S3 [×7], C6 [×3], C6 [×7], C2×C4 [×3], C2×C4 [×9], D4 [×24], C23, C23 [×3], C23 [×91], Dic3 [×3], C12 [×3], D6 [×12], D6 [×61], C2×C6, C2×C6 [×6], C2×C6 [×17], C22⋊C4 [×3], C22⋊C4 [×9], C22×C4 [×3], C2×D4 [×3], C2×D4 [×21], C24, C24 [×19], C4×S3 [×6], D12 [×6], C2×Dic3 [×3], C3⋊D4 [×12], C2×C12 [×3], C3×D4 [×6], C22×S3 [×2], C22×S3 [×15], C22×S3 [×68], C22×C6, C22×C6 [×3], C22×C6 [×6], C2×C22⋊C4 [×3], C22≀C2, C22≀C2 [×7], C22×D4 [×3], C25, D6⋊C4 [×6], C6.D4 [×3], C3×C22⋊C4 [×3], S3×C2×C4 [×3], C2×D12 [×3], S3×D4 [×12], C2×C3⋊D4 [×6], C6×D4 [×3], S3×C23, S3×C23 [×6], S3×C23 [×12], C23×C6, C2×C22≀C2, S3×C22⋊C4 [×3], D6⋊D4 [×3], C232D6 [×3], C244S3, C3×C22≀C2, C2×S3×D4 [×3], S3×C24, S3×C22≀C2
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C22×S3 [×7], C22≀C2 [×4], C22×D4 [×3], S3×D4 [×6], S3×C23, C2×C22≀C2, C2×S3×D4 [×3], S3×C22≀C2

Permutation representations of S3×C22≀C2
On 24 points - transitive group 24T360
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 10)(2 12)(3 11)(4 7)(5 9)(6 8)(13 22)(14 24)(15 23)(16 19)(17 21)(18 20)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)

G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10)(2,12)(3,11)(4,7)(5,9)(6,8)(13,22)(14,24)(15,23)(16,19)(17,21)(18,20), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10)(2,12)(3,11)(4,7)(5,9)(6,8)(13,22)(14,24)(15,23)(16,19)(17,21)(18,20), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,10),(2,12),(3,11),(4,7),(5,9),(6,8),(13,22),(14,24),(15,23),(16,19),(17,21),(18,20)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)])

G:=TransitiveGroup(24,360);

42 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 2N 2O ··· 2T 2U 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 6J 12A 12B 12C order 1 2 2 2 2 ··· 2 2 2 2 2 2 2 ··· 2 2 3 4 4 4 4 4 4 6 6 6 6 ··· 6 6 12 12 12 size 1 1 1 1 2 ··· 2 3 3 3 3 4 6 ··· 6 12 2 4 4 4 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 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 S3×D4 kernel S3×C22≀C2 S3×C22⋊C4 D6⋊D4 C23⋊2D6 C24⋊4S3 C3×C22≀C2 C2×S3×D4 S3×C24 C22≀C2 C22×S3 C22⋊C4 C2×D4 C24 C22 # reps 1 3 3 3 1 1 3 1 1 12 3 3 1 6

Matrix representation of S3×C22≀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 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 5 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 0 0 12 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 1
,
 12 0 0 0 0 0 0 12 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
,
 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
,
 1 3 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

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S3×C22≀C2 in GAP, Magma, Sage, TeX

S_3\times C_2^2\wr C_2
% in TeX

G:=Group("S3xC2^2wrC2");
// GroupNames label

G:=SmallGroup(192,1147);
// by ID

G=gap.SmallGroup(192,1147);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,570,185,6278]);
// Polycyclic

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

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