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G = C8⋊S4order 192 = 26·3

3rd semidirect product of C8 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C83S4, A41M4(2), A4⋊C4.C4, (C2×S4).C4, A4⋊C84C2, (C8×A4)⋊5C2, C2.7(C4×S4), (C4×S4).2C2, C4.27(C2×S4), (C22×C8)⋊4S3, C22⋊(C8⋊S3), C23.2(C4×S3), (C22×C4).9D6, (C4×A4).13C22, (C2×A4).2(C2×C4), SmallGroup(192,959)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C8⋊S4
C1C22A4C2×A4C4×A4C4×S4 — C8⋊S4
A4C2×A4 — C8⋊S4
C1C4C8

Generators and relations for C8⋊S4
 G = < a,b,c,d,e | a8=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a5, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 246 in 72 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×4], C22, C22 [×5], S3, C6, C8, C8 [×3], C2×C4 [×7], D4 [×2], C23, C23, Dic3, C12, A4, D6, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, S4, C2×A4, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C8⋊S3, A4⋊C4, C4×A4, C2×S4, C89D4, A4⋊C8, C8×A4, C4×S4, C8⋊S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, M4(2), C4×S3, S4, C8⋊S3, C2×S4, C4×S4, C8⋊S4

Character table of C8⋊S4

 class 12A2B2C2D34A4B4C4D4E4F4G68A8B8C8D8E8F8G8H12A12B24A24B24C24D
 size 113312811331212128226612121212888888
ρ11111111111111111111111111111    trivial
ρ21111-111111-1-1-111111-1-1-1-1111111    linear of order 2
ρ31111-111111-1-1-11-1-1-1-1111111-1-1-1-1    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ51111-11-1-1-1-11-111i-ii-ii-ii-i-1-1i-ii-i    linear of order 4
ρ6111111-1-1-1-1-11-11-ii-iii-ii-i-1-1-ii-ii    linear of order 4
ρ7111111-1-1-1-1-11-11i-ii-i-ii-ii-1-1i-ii-i    linear of order 4
ρ81111-11-1-1-1-11-111-ii-ii-ii-ii-1-1-ii-ii    linear of order 4
ρ922220-12222000-1-2-2-2-20000-1-11111    orthogonal lifted from D6
ρ1022220-12222000-122220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ112-2-22022i-2i2i-2i000-200000000-2i2i0000    complex lifted from M4(2)
ρ1222220-1-2-2-2-2000-1-2i2i-2i2i000011i-ii-i    complex lifted from C4×S3
ρ1322220-1-2-2-2-2000-12i-2i2i-2i000011-ii-ii    complex lifted from C4×S3
ρ142-2-2202-2i2i-2i2i000-2000000002i-2i0000    complex lifted from M4(2)
ρ152-2-220-1-2i2i-2i2i000100000000-ii8ζ3883ζ38385ζ38587ζ387    complex lifted from C8⋊S3
ρ162-2-220-12i-2i2i-2i000100000000i-i87ζ38785ζ38583ζ3838ζ38    complex lifted from C8⋊S3
ρ172-2-220-1-2i2i-2i2i000100000000-ii85ζ38587ζ3878ζ3883ζ383    complex lifted from C8⋊S3
ρ182-2-220-12i-2i2i-2i000100000000i-i83ζ3838ζ3887ζ38785ζ385    complex lifted from C8⋊S3
ρ1933-1-1-1033-1-111-10-3-311-1-111000000    orthogonal lifted from C2×S4
ρ2033-1-11033-1-1-1-110-3-31111-1-1000000    orthogonal lifted from C2×S4
ρ2133-1-11033-1-1-1-11033-1-1-1-111000000    orthogonal lifted from S4
ρ2233-1-1-1033-1-111-1033-1-111-1-1000000    orthogonal lifted from S4
ρ2333-1-1-10-3-311-1110-3i3ii-ii-i-ii000000    complex lifted from C4×S4
ρ2433-1-1-10-3-311-11103i-3i-ii-iii-i000000    complex lifted from C4×S4
ρ2533-1-110-3-3111-1-10-3i3ii-i-iii-i000000    complex lifted from C4×S4
ρ2633-1-110-3-3111-1-103i-3i-iii-i-ii000000    complex lifted from C4×S4
ρ276-62-200-6i6i2i-2i000000000000000000    complex faithful
ρ286-62-2006i-6i-2i2i000000000000000000    complex faithful

Permutation representations of C8⋊S4
On 24 points - transitive group 24T323
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 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 11 19)(2 12 20)(3 13 21)(4 14 22)(5 15 23)(6 16 24)(7 9 17)(8 10 18)
(2 6)(4 8)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)

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

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

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

G:=TransitiveGroup(24,323);

Matrix representation of C8⋊S4 in GL5(𝔽73)

4513000
2628000
004600
000460
000046
,
10000
01000
00001
00727272
00100
,
10000
01000
00010
00100
00727272
,
10000
01000
00010
00727272
00001
,
7254000
01000
00010
00100
00001

G:=sub<GL(5,GF(73))| [45,26,0,0,0,13,28,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,1,0,72,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,72,1],[72,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C8⋊S4 in GAP, Magma, Sage, TeX

C_8\rtimes S_4
% in TeX

G:=Group("C8:S4");
// GroupNames label

G:=SmallGroup(192,959);
// by ID

G=gap.SmallGroup(192,959);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,141,36,58,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of C8⋊S4 in TeX

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