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G = A4⋊D8  order 192 = 26·3

The semidirect product of A4 and D8 acting via D8/C8=C2

non-abelian, soluble, monomial

Aliases: C8⋊1S4, A4⋊1D8, C22⋊D24, C23.10D12, C4⋊S4⋊1C2, (C8×A4)⋊1C2, C2.8(C4⋊S4), C4.18(C2×S4), (C22×C8)⋊2S3, (C2×A4).3D4, (C22×C4).15D6, (C4×A4).10C22, SmallGroup(192,961)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C4×A4 — A4⋊D8
Chief series
C1 — C22 — A4 — C2×A4 — C4×A4 — C4⋊S4 — A4⋊D8
Lower central
A4 — C2×A4 — C4×A4 — A4⋊D8
Upper central
C1 — C2 — C4 — C8

Generators and relations for A4⋊D8
 G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=eae=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 442 in 79 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, D4, C23, C23, C12, A4, D6, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, D12, S4, C2×A4, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, D24, C4×A4, C2×S4, C8⋊7D4, C8×A4, C4⋊S4, A4⋊D8
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, S4, D24, C2×S4, C4⋊S4, A4⋊D8

Character table of A4⋊D8

 class 12A2B2C2D2E34A4B4C4D68A8B8C8D12A12B24A24B24C24D
 size 11332424826242482266888888
ρ11111111111111111111111    trivial
ρ21111-111111-11-1-1-1-111-1-1-1-1    linear of order 2
ρ31111-1-1111-1-111111111111    linear of order 2
ρ411111-1111-111-1-1-1-111-1-1-1-1    linear of order 2
ρ52222002-2-20020000-2-20000    orthogonal lifted from D4
ρ6222200-12200-12222-1-1-1-1-1-1    orthogonal lifted from S3
ρ7222200-12200-1-2-2-2-2-1-11111    orthogonal lifted from D6
ρ82-2-220020000-2-√2√2√2-√200-√2-√2√2√2    orthogonal lifted from D8
ρ92-2-220020000-2√2-√2-√2√200√2√2-√2-√2    orthogonal lifted from D8
ρ10222200-1-2-200-1000011√3-√3√3-√3    orthogonal lifted from D12
ρ11222200-1-2-200-1000011-√3√3-√3√3    orthogonal lifted from D12
ρ122-2-2200-100001-√2√2√2-√2√3-√3ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85    orthogonal lifted from D24
ρ132-2-2200-100001√2-√2-√2√2-√3√3ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8    orthogonal lifted from D24
ρ142-2-2200-100001-√2√2√2-√2-√3√3ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32    orthogonal lifted from D24
ρ152-2-2200-100001√2-√2-√2√2√3-√3ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3    orthogonal lifted from D24
ρ1633-1-11103-1-1-1033-1-1000000    orthogonal lifted from S4
ρ1733-1-11-103-11-10-3-311000000    orthogonal lifted from C2×S4
ρ1833-1-1-1-103-111033-1-1000000    orthogonal lifted from S4
ρ1933-1-1-1103-1-110-3-311000000    orthogonal lifted from C2×S4
ρ2066-2-2000-620000000000000    orthogonal lifted from C4⋊S4
ρ216-62-2000000003√2-3√2√2-√2000000    orthogonal faithful
ρ226-62-200000000-3√23√2-√2√2000000    orthogonal faithful

Permutation representations of A4⋊D8
►On 24 points - transitive group 24T324
Generators in S24
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)

(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)

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

(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 7)(2 6)(3 5)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)
 
G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,22,11)(2,23,12)(3,24,13)(4,17,14)(5,18,15)(6,19,16)(7,20,9)(8,21,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,22,11)(2,23,12)(3,24,13)(4,17,14)(5,18,15)(6,19,16)(7,20,9)(8,21,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,7)(2,6)(3,5)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23) );
 
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,22,11),(2,23,12),(3,24,13),(4,17,14),(5,18,15),(6,19,16),(7,20,9),(8,21,10)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,7),(2,6),(3,5),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,24),(16,23)]])
 
G:=TransitiveGroup(24,324);
 

Matrix representation of A4⋊D8 ►in GL5(𝔽73)

10000
01000
007200
007201
007210
,
10000
01000
000172
001072
000072
,
10000
01000
000720
001720
000721
,
3272000
10000
00100
00010
00001
,
3272000
141000
00010
00100
00001

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72,72,0,0,0,0,1],[32,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[32,1,0,0,0,72,41,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
 

A4⋊D8 in GAP, Magma, Sage, TeX 

A_4\rtimes D_8
 
% in TeX
 
G:=Group("A4:D8");
 
// GroupNames label
 
G:=SmallGroup(192,961);
 
// by ID
 
G=gap.SmallGroup(192,961);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,92,254,58,1124,4037,285,2358,475]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=e*a*e=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of A4⋊D8 in TeX

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