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## G = A4⋊D8order 192 = 26·3

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

Aliases: C81S4, A41D8, C22⋊D24, C23.10D12, C4⋊S41C2, (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 [×4], C3, C4, C4 [×3], C22, C22 [×8], S3 [×2], C6, C8, C8, C2×C4 [×4], D4 [×8], C23, C23 [×2], C12, A4, D6 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×2], C22×C4, C2×D4 [×4], C24, D12 [×2], S4 [×2], C2×A4, D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, D24, C4×A4, C2×S4 [×2], C87D4, C8×A4, C4⋊S4 [×2], A4⋊D8
Quotients: C1, C2 [×3], C22, S3, D4, D6, D8, D12, S4, D24, C2×S4, C4⋊S4, A4⋊D8

Character table of A4⋊D8

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6 8A 8B 8C 8D 12A 12B 24A 24B 24C 24D size 1 1 3 3 24 24 8 2 6 24 24 8 2 2 6 6 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 trivial ρ2 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 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 linear of order 2 ρ5 2 2 2 2 0 0 2 -2 -2 0 0 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 2 0 0 -1 2 2 0 0 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 0 0 -1 2 2 0 0 -1 -2 -2 -2 -2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ8 2 -2 -2 2 0 0 2 0 0 0 0 -2 -√2 √2 √2 -√2 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ9 2 -2 -2 2 0 0 2 0 0 0 0 -2 √2 -√2 -√2 √2 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ10 2 2 2 2 0 0 -1 -2 -2 0 0 -1 0 0 0 0 1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ11 2 2 2 2 0 0 -1 -2 -2 0 0 -1 0 0 0 0 1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ12 2 -2 -2 2 0 0 -1 0 0 0 0 1 -√2 √2 √2 -√2 √3 -√3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 orthogonal lifted from D24 ρ13 2 -2 -2 2 0 0 -1 0 0 0 0 1 √2 -√2 -√2 √2 -√3 √3 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 orthogonal lifted from D24 ρ14 2 -2 -2 2 0 0 -1 0 0 0 0 1 -√2 √2 √2 -√2 -√3 √3 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 orthogonal lifted from D24 ρ15 2 -2 -2 2 0 0 -1 0 0 0 0 1 √2 -√2 -√2 √2 √3 -√3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 orthogonal lifted from D24 ρ16 3 3 -1 -1 1 1 0 3 -1 -1 -1 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ17 3 3 -1 -1 1 -1 0 3 -1 1 -1 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 -1 -1 -1 -1 0 3 -1 1 1 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ19 3 3 -1 -1 -1 1 0 3 -1 -1 1 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ20 6 6 -2 -2 0 0 0 -6 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4 ρ21 6 -6 2 -2 0 0 0 0 0 0 0 0 3√2 -3√2 √2 -√2 0 0 0 0 0 0 orthogonal faithful ρ22 6 -6 2 -2 0 0 0 0 0 0 0 0 -3√2 3√2 -√2 √2 0 0 0 0 0 0 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 10)(2 23 11)(3 24 12)(4 17 13)(5 18 14)(6 19 15)(7 20 16)(8 21 9)
(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 21)(10 20)(11 19)(12 18)(13 17)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,324);

Matrix representation of A4⋊D8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 72 0 1 0 0 72 1 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
,
 32 72 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 72 0 0 0 1 41 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

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

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