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## G = S3×C23⋊C4order 192 = 26·3

### Direct product of S3 and C23⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C23⋊C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 768 in 210 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2 [×10], C3, C4 [×6], C22, C22 [×2], C22 [×22], S3 [×2], S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×11], D4 [×8], C23 [×2], C23 [×13], Dic3 [×3], C12 [×3], D6 [×2], D6 [×2], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4, C2×D4 [×7], C24 [×2], C4×S3 [×6], D12 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×S3 [×4], C22×S3 [×6], C22×C6 [×2], C23⋊C4, C23⋊C4 [×3], C2×C22⋊C4 [×2], C22×D4, D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, S3×D4 [×4], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C2×C23⋊C4, C23.6D6 [×2], C23.7D6, C3×C23⋊C4, S3×C22⋊C4 [×2], C2×S3×D4, S3×C23⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C23⋊C4 [×2], C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C2×C23⋊C4, S3×C22⋊C4, S3×C23⋊C4

Permutation representations of S3×C23⋊C4
On 24 points - transitive group 24T337
Generators in S24
(1 12 10)(2 11 9)(3 7 6)(4 8 5)(13 24 19)(14 21 20)(15 22 17)(16 23 18)
(1 6)(2 5)(3 10)(4 9)(7 12)(8 11)(13 22)(14 23)(15 24)(16 21)(17 19)(18 20)
(1 13)(2 14)(3 17)(4 18)(5 23)(6 22)(7 15)(8 16)(9 20)(10 19)(11 21)(12 24)
(1 7)(3 10)(6 12)(13 15)(17 19)(22 24)
(1 7)(2 8)(3 10)(4 9)(5 11)(6 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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)

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

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

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

G:=TransitiveGroup(24,337);

On 24 points - transitive group 24T341
Generators in S24
(1 6 9)(2 7 10)(3 8 11)(4 5 12)(13 22 17)(14 23 18)(15 24 19)(16 21 20)
(1 14)(2 15)(3 16)(4 13)(5 17)(6 18)(7 19)(8 20)(9 23)(10 24)(11 21)(12 22)
(2 15)(3 16)(7 24)(8 21)(10 19)(11 20)
(2 15)(4 13)(5 22)(7 24)(10 19)(12 17)
(1 14)(2 15)(3 16)(4 13)(5 22)(6 23)(7 24)(8 21)(9 18)(10 19)(11 20)(12 17)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

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

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

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

G:=TransitiveGroup(24,341);

On 24 points - transitive group 24T363
Generators in S24
(1 19 21)(2 20 22)(3 17 23)(4 18 24)(5 14 9)(6 15 10)(7 16 11)(8 13 12)
(1 14)(2 15)(3 16)(4 13)(5 19)(6 20)(7 17)(8 18)(9 21)(10 22)(11 23)(12 24)
(1 4)(2 16)(3 15)(5 8)(6 23)(7 22)(9 12)(10 17)(11 20)(13 14)(18 19)(21 24)
(1 16)(2 4)(3 14)(5 23)(6 8)(7 21)(9 17)(10 12)(11 19)(13 15)(18 20)(22 24)
(1 14)(2 15)(3 16)(4 13)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

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

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

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

G:=TransitiveGroup(24,363);

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A ··· 4E 4F ··· 4J 6A 6B 6C 6D 6E 12A ··· 12E order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 6 6 12 ··· 12 size 1 1 2 2 2 3 3 4 6 6 6 12 2 4 ··· 4 12 ··· 12 2 4 4 4 8 8 ··· 8

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D6 D6 C4×S3 C4×S3 C23⋊C4 S3×D4 S3×C23⋊C4 kernel S3×C23⋊C4 C23.6D6 C23.7D6 C3×C23⋊C4 S3×C22⋊C4 C2×S3×D4 S3×C2×C4 C2×D12 C2×C3⋊D4 S3×C23 C23⋊C4 C22×S3 C22⋊C4 C2×D4 C2×C4 C23 S3 C22 C1 # reps 1 2 1 1 2 1 2 2 2 2 1 4 2 1 2 2 2 2 1

Matrix representation of S3×C23⋊C4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 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 1
,
 12 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 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 1 1 12 8 0 0 1 0 0 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 12 0 0 0 0 0 0 1 0 0 0 3 0 10 12
,
 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 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 12 8 0 0 0 0 3 1

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

S3×C23⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_2^3\rtimes C_4
% in TeX

G:=Group("S3xC2^3:C4");
// GroupNames label

G:=SmallGroup(192,302);
// by ID

G=gap.SmallGroup(192,302);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,570,438,6278]);
// Polycyclic

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

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