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## G = A4×D8order 192 = 26·3

### Direct product of A4 and D8

Aliases: A4×D8, D4⋊(C2×A4), C81(C2×A4), C22⋊(C3×D8), (C22×D8)⋊C3, (C8×A4)⋊3C2, (D4×A4)⋊4C2, (C22×D4)⋊C6, C2.6(D4×A4), (C22×C8)⋊1C6, (C2×A4).14D4, C4.1(C22×A4), C23.23(C3×D4), (C4×A4).17C22, (C22×C4).1(C2×C6), SmallGroup(192,1014)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C4 — A4×D8
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — D4×A4 — A4×D8
 Lower central C22 — C23 — C22×C4 — A4×D8
 Upper central C1 — C2 — C4 — D8

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

Subgroups: 424 in 93 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22, C22 [×14], C6 [×3], C8, C8, C2×C4 [×2], D4 [×2], D4 [×6], C23, C23 [×8], C12, A4, C2×C6 [×2], C2×C8 [×2], D8, D8 [×5], C22×C4, C2×D4 [×6], C24 [×2], C24, C3×D4 [×2], C2×A4, C2×A4 [×2], C22×C8, C2×D8 [×4], C22×D4 [×2], C3×D8, C4×A4, C22×A4 [×2], C22×D8, C8×A4, D4×A4 [×2], A4×D8
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, D8, C3×D4, C2×A4 [×3], C3×D8, C22×A4, D4×A4, A4×D8

Character table of A4×D8

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 24A 24B 24C 24D size 1 1 3 3 4 4 12 12 4 4 2 6 4 4 16 16 16 16 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 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 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 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 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ7 1 1 1 1 -1 1 1 -1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ8 1 1 1 1 -1 1 1 -1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ9 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ10 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ11 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ12 1 1 1 1 1 -1 -1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ13 2 2 2 2 0 0 0 0 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 -2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 √2 -√2 √2 -√2 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 -2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 -√2 √2 -√2 √2 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 2 2 0 0 0 0 -1-√-3 -1+√-3 -2 -2 -1+√-3 -1-√-3 0 0 0 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 complex lifted from C3×D4 ρ17 2 2 2 2 0 0 0 0 -1+√-3 -1-√-3 -2 -2 -1-√-3 -1+√-3 0 0 0 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 complex lifted from C3×D4 ρ18 2 -2 -2 2 0 0 0 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 0 0 -√2 √2 -√2 √2 0 0 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 complex lifted from C3×D8 ρ19 2 -2 -2 2 0 0 0 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 0 0 √2 -√2 √2 -√2 0 0 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 complex lifted from C3×D8 ρ20 2 -2 -2 2 0 0 0 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 0 0 -√2 √2 -√2 √2 0 0 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 complex lifted from C3×D8 ρ21 2 -2 -2 2 0 0 0 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 0 0 √2 -√2 √2 -√2 0 0 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 complex lifted from C3×D8 ρ22 3 3 -1 -1 -3 -3 1 1 0 0 3 -1 0 0 0 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ23 3 3 -1 -1 -3 3 -1 1 0 0 3 -1 0 0 0 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ24 3 3 -1 -1 3 3 -1 -1 0 0 3 -1 0 0 0 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ25 3 3 -1 -1 3 -3 1 -1 0 0 3 -1 0 0 0 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ26 6 6 -2 -2 0 0 0 0 0 0 -6 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4×A4 ρ27 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3√2 3√2 √2 -√2 0 0 0 0 0 0 orthogonal faithful ρ28 6 -6 2 -2 0 0 0 0 0 0 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 24T328
Generators in S24
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 18)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)

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

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

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

G:=TransitiveGroup(24,328);

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 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 0 1 0 0 1 0 0 0 0 0 1 0
,
 72 1 0 0 0 39 33 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 39 1 0 0 0 0 0 1 0 0 0 0 0 1 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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[72,39,0,0,0,1,33,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,39,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D8 in GAP, Magma, Sage, TeX

A_4\times D_8
% in TeX

G:=Group("A4xD8");
// GroupNames label

G:=SmallGroup(192,1014);
// by ID

G=gap.SmallGroup(192,1014);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,197,1011,514,80,1027,1784]);
// Polycyclic

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

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