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## G = C2×C4⋊S4order 192 = 26·3

### Direct product of C2 and C4⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C2×C4⋊S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C22×S4 — C2×C4⋊S4
 Lower central A4 — C2×A4 — C2×C4⋊S4
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C4⋊S4
G = < a,b,c,d,e,f | a2=b4=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 998 in 233 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×6], C22 [×2], C22 [×30], S3 [×4], C6 [×3], C2×C4, C2×C4 [×16], D4 [×24], C23, C23 [×2], C23 [×20], C12 [×2], A4, D6 [×8], C2×C6, C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×6], C2×D4 [×24], C24, C24 [×2], D12 [×4], C2×C12, S4 [×4], C2×A4, C2×A4 [×2], C22×S3 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], C4×A4 [×2], C2×D12, C2×S4 [×4], C2×S4 [×4], C22×A4, C2×C4⋊D4, C4⋊S4 [×4], C2×C4×A4, C22×S4 [×2], C2×C4⋊S4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], S4, C22×S3, C2×D12, C2×S4 [×3], C4⋊S4 [×2], C22×S4, C2×C4⋊S4

Character table of C2×C4⋊S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 3 3 3 3 12 12 12 12 8 2 2 6 6 12 12 12 12 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 2 2 -2 -2 -2 -2 2 2 0 0 0 0 -1 -2 2 2 -2 0 0 0 0 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 2 2 2 2 0 0 0 0 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 -2 -2 2 2 -2 -2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 2 -2 -2 2 -2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 -2 -2 2 2 0 0 0 0 -1 2 -2 -2 2 0 0 0 0 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ14 2 2 2 2 2 2 2 2 0 0 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 -2 2 -2 -2 2 -2 2 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ16 2 -2 2 -2 -2 2 -2 2 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ17 2 -2 -2 2 2 -2 -2 2 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ18 2 -2 -2 2 2 -2 -2 2 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ19 3 3 3 3 -1 -1 -1 -1 1 1 1 1 0 3 3 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 3 3 -1 -1 1 1 1 1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ21 3 3 3 3 -1 -1 -1 -1 -1 1 -1 1 0 -3 -3 1 1 1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 3 3 -1 -1 -1 -1 1 -1 1 -1 0 -3 -3 1 1 -1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 3 -3 -3 1 1 -1 -1 1 1 -1 -1 0 -3 3 -1 1 -1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ24 3 3 -3 -3 1 1 -1 -1 -1 -1 1 1 0 -3 3 -1 1 1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ25 3 3 -3 -3 1 1 -1 -1 -1 1 1 -1 0 3 -3 1 -1 1 -1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 3 3 -3 -3 1 1 -1 -1 1 -1 -1 1 0 3 -3 1 -1 -1 1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ27 6 -6 6 -6 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4 ρ28 6 -6 -6 6 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4

Permutation representations of C2×C4⋊S4
On 24 points - transitive group 24T394
Generators in S24
(1 14)(2 15)(3 16)(4 13)(5 12)(6 9)(7 10)(8 11)(17 24)(18 21)(19 22)(20 23)
(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 16)(2 13)(3 14)(4 15)(17 22)(18 23)(19 24)(20 21)
(5 10)(6 11)(7 12)(8 9)(17 22)(18 23)(19 24)(20 21)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 13 19)(6 14 20)(7 15 17)(8 16 18)
(1 16)(2 15)(3 14)(4 13)(5 22)(6 21)(7 24)(8 23)(9 18)(10 17)(11 20)(12 19)

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

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

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

G:=TransitiveGroup(24,394);

On 24 points - transitive group 24T419
Generators in S24
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(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)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 24)
(1 22)(2 21)(3 24)(4 23)(5 12)(6 11)(7 10)(8 9)(13 19)(14 18)(15 17)(16 20)

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

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

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

G:=TransitiveGroup(24,419);

Matrix representation of C2×C4⋊S4 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 9 6 0 0 0 0 0 8 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 1
,
 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 11 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,8,0,0,0,0,0,6,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,11,1,12],[12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,12,0,0,0,0,0,0,12] >;

C2×C4⋊S4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes S_4
% in TeX

G:=Group("C2xC4:S4");
// GroupNames label

G:=SmallGroup(192,1470);
// by ID

G=gap.SmallGroup(192,1470);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,254,58,1124,4037,285,2358,475]);
// Polycyclic

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

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