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G = C2×C4⋊S4order 192 = 26·3

Direct product of C2 and C4⋊S4

direct product, non-abelian, soluble, monomial

Aliases: C2×C4⋊S4, C232D12, C24.9D6, C42(C2×S4), (C2×C4)⋊2S4, A41(C2×D4), (C2×A4)⋊1D4, C22⋊(C2×D12), (C23×C4)⋊3S3, (C22×C4)⋊3D6, (C4×A4)⋊2C22, (C2×S4)⋊1C22, (C22×S4)⋊1C2, C2.4(C22×S4), (C2×A4).3C23, C22.24(C2×S4), C23.3(C22×S3), (C22×A4).10C22, (C2×C4×A4)⋊3C2, SmallGroup(192,1470)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C2×C4⋊S4
C1C22A4C2×A4C2×S4C22×S4 — C2×C4⋊S4
A4C2×A4 — C2×C4⋊S4
C1C22C2×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 12A2B2C2D2E2F2G2H2I2J2K34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D
 size 111133331212121282266121212128888888
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-111111-1-1-1-11111111    linear of order 2
ρ311-1-1-1-11111-1-11-111-111-1-11-1-111-1-1    linear of order 2
ρ411-1-1-1-111-1-1111-111-1-1-1111-1-111-1-1    linear of order 2
ρ5111111111-11-11-1-1-1-11-11-1111-1-1-1-1    linear of order 2
ρ611111111-11-111-1-1-1-1-11-11111-1-1-1-1    linear of order 2
ρ711-1-1-1-1111-1-1111-1-111-1-111-1-1-1-111    linear of order 2
ρ811-1-1-1-111-111-111-1-11-111-11-1-1-1-111    linear of order 2
ρ922-2-2-2-2220000-1-222-20000-111-1-111    orthogonal lifted from D6
ρ10222222220000-1-2-2-2-20000-1-1-11111    orthogonal lifted from D6
ρ112-2-222-2-220000200000000-22-20000    orthogonal lifted from D4
ρ122-22-2-22-220000200000000-2-220000    orthogonal lifted from D4
ρ1322-2-2-2-2220000-12-2-220000-11111-1-1    orthogonal lifted from D6
ρ14222222220000-122220000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ152-22-2-22-220000-10000000011-1-333-3    orthogonal lifted from D12
ρ162-22-2-22-220000-10000000011-13-3-33    orthogonal lifted from D12
ρ172-2-222-2-220000-1000000001-113-33-3    orthogonal lifted from D12
ρ182-2-222-2-220000-1000000001-11-33-33    orthogonal lifted from D12
ρ193333-1-1-1-11111033-1-1-1-1-1-10000000    orthogonal lifted from S4
ρ203333-1-1-1-1-1-1-1-1033-1-111110000000    orthogonal lifted from S4
ρ213333-1-1-1-1-11-110-3-3111-11-10000000    orthogonal lifted from C2×S4
ρ223333-1-1-1-11-11-10-3-311-11-110000000    orthogonal lifted from C2×S4
ρ2333-3-311-1-111-1-10-33-11-1-1110000000    orthogonal lifted from C2×S4
ρ2433-3-311-1-1-1-1110-33-1111-1-10000000    orthogonal lifted from C2×S4
ρ2533-3-311-1-1-111-103-31-11-1-110000000    orthogonal lifted from C2×S4
ρ2633-3-311-1-11-1-1103-31-1-111-10000000    orthogonal lifted from C2×S4
ρ276-66-62-22-200000000000000000000    orthogonal lifted from C4⋊S4
ρ286-6-66-222-200000000000000000000    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)

12000000
01200000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0096000
0084000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
00001200
0000110
00000012
,
1000000
0100000
0010000
0001000
00001200
00000120
00001201
,
0100000
121200000
0010000
0001000
00001011
0000001
000001212
,
12000000
1100000
00120000
0031000
000012110
0000010
000001212

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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Character table of C2×C4⋊S4 in TeX

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