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G = C82S4order 192 = 26·3

2nd semidirect product of C8 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C82S4, A41SD16, C23.9D12, (C8×A4)⋊2C2, C4⋊S4.1C2, A4⋊Q81C2, C2.7(C4⋊S4), C4.17(C2×S4), (C22×C8)⋊3S3, (C2×A4).2D4, C22⋊(C24⋊C2), (C4×A4).9C22, (C22×C4).14D6, SmallGroup(192,960)

Series: Derived Chief Lower central Upper central

C1C22C4×A4 — C82S4
C1C22A4C2×A4C4×A4C4⋊S4 — C82S4
A4C2×A4C4×A4 — C82S4
C1C2C4C8

Generators and relations for C82S4
 G = < a,b,c,d,e | a8=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a3, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 350 in 72 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×4], C22, C22 [×5], S3, C6, C8, C8, C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, Dic3, C12, A4, D6, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, C24, Dic6, D12, S4, C2×A4, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C24⋊C2, A4⋊C4, C4×A4, C2×S4, C88D4, C8×A4, A4⋊Q8, C4⋊S4, C82S4
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, D12, S4, C24⋊C2, C2×S4, C4⋊S4, C82S4

Character table of C82S4

 class 12A2B2C2D34A4B4C4D4E68A8B8C8D12A12B24A24B24C24D
 size 11332482624242482266888888
ρ11111111111111111111111    trivial
ρ21111-111111-11-1-1-1-111-1-1-1-1    linear of order 2
ρ31111-1111-1-1-111111111111    linear of order 2
ρ411111111-1-111-1-1-1-111-1-1-1-1    linear of order 2
ρ5222202-2-200020000-2-20000    orthogonal lifted from D4
ρ622220-122000-12222-1-1-1-1-1-1    orthogonal lifted from S3
ρ722220-122000-1-2-2-2-2-1-11111    orthogonal lifted from D6
ρ822220-1-2-2000-10000113-33-3    orthogonal lifted from D12
ρ922220-1-2-2000-1000011-33-33    orthogonal lifted from D12
ρ102-2-220200000-2--2-2-2--200--2--2-2-2    complex lifted from SD16
ρ112-2-220200000-2-2--2--2-200-2-2--2--2    complex lifted from SD16
ρ122-2-220-1000001--2-2-2--2-3383ζ38ζ3883ζ328ζ32887ζ385ζ38587ζ3285ζ3285    complex lifted from C24⋊C2
ρ132-2-220-1000001-2--2--2-2-3387ζ385ζ38587ζ3285ζ328583ζ38ζ3883ζ328ζ328    complex lifted from C24⋊C2
ρ142-2-220-1000001--2-2-2--23-383ζ328ζ32883ζ38ζ3887ζ3285ζ328587ζ385ζ385    complex lifted from C24⋊C2
ρ152-2-220-1000001-2--2--2-23-387ζ3285ζ328587ζ385ζ38583ζ328ζ32883ζ38ζ38    complex lifted from C24⋊C2
ρ1633-1-1-103-11-11033-1-1000000    orthogonal lifted from S4
ρ1733-1-1-103-1-1110-3-311000000    orthogonal lifted from C2×S4
ρ1833-1-1103-11-1-10-3-311000000    orthogonal lifted from C2×S4
ρ1933-1-1103-1-11-1033-1-1000000    orthogonal lifted from S4
ρ2066-2-200-6200000000000000    orthogonal lifted from C4⋊S4
ρ216-62-200000000-3-23-2--2-2000000    complex faithful
ρ226-62-2000000003-2-3-2-2--2000000    complex faithful

Permutation representations of C82S4
On 24 points - transitive group 24T325
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 4)(3 7)(6 8)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)

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

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

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

G:=TransitiveGroup(24,325);

Matrix representation of C82S4 in GL5(𝔽73)

6167000
120000
00100
00010
00001
,
10000
01000
000721
000720
001720
,
10000
01000
000172
001072
000072
,
10000
01000
007210
007200
007201
,
7272000
01000
00010
00100
00001

G:=sub<GL(5,GF(73))| [61,12,0,0,0,67,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C82S4 in GAP, Magma, Sage, TeX

C_8\rtimes_2S_4
% in TeX

G:=Group("C8:2S4");
// GroupNames label

G:=SmallGroup(192,960);
// by ID

G=gap.SmallGroup(192,960);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,36,254,58,1124,4037,285,2358,475]);
// 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^3,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

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Character table of C82S4 in TeX

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