Copied to
clipboard

G = A4×D8  order 192 = 26·3

Direct product of A4 and D8

direct product, metabelian, soluble, monomial

Aliases: A4×D8, D4⋊(C2×A4), C8⋊1(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, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, D4, D4, C23, C23, C12, A4, C2×C6, C2×C8, D8, D8, C22×C4, C2×D4, C24, C24, C3×D4, C2×A4, C2×A4, C22×C8, C2×D8, C22×D4, C3×D8, C4×A4, C22×A4, C22×D8, C8×A4, D4×A4, A4×D8
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, D8, C3×D4, C2×A4, C3×D8, C22×A4, D4×A4, A4×D8

Character table of A4×D8

 class 12A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F8A8B8C8D12A12B24A24B24C24D
 size 1133441212442644161616162266888888
ρ11111111111111111111111111111    trivial
ρ211111-1-1111111111-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ31111-111-1111111-1-111-1-1-1-111-1-1-1-1    linear of order 2
ρ41111-1-1-1-1111111-1-1-1-11111111111    linear of order 2
ρ511111-1-11ζ3ζ3211ζ32ζ3ζ3ζ32ζ65ζ6-1-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ61111-1-1-1-1ζ3ζ3211ζ32ζ3ζ65ζ6ζ65ζ61111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 6
ρ71111-111-1ζ3ζ3211ζ32ζ3ζ65ζ6ζ3ζ32-1-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ81111-111-1ζ32ζ311ζ3ζ32ζ6ζ65ζ32ζ3-1-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ91111-1-1-1-1ζ32ζ311ζ3ζ32ζ6ζ65ζ6ζ651111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 6
ρ1011111111ζ3ζ3211ζ32ζ3ζ3ζ32ζ3ζ321111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ1111111111ζ32ζ311ζ3ζ32ζ32ζ3ζ32ζ31111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ1211111-1-11ζ32ζ311ζ3ζ32ζ32ζ3ζ6ζ65-1-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ132222000022-2-22200000000-2-20000    orthogonal lifted from D4
ρ142-2-2200002200-2-20000√2-√2√2-√200√2-√2√2-√2    orthogonal lifted from D8
ρ152-2-2200002200-2-20000-√2√2-√2√200-√2√2-√2√2    orthogonal lifted from D8
ρ1622220000-1-√-3-1+√-3-2-2-1+√-3-1-√-3000000001+√-31-√-30000    complex lifted from C3×D4
ρ1722220000-1+√-3-1-√-3-2-2-1-√-3-1+√-3000000001-√-31+√-30000    complex lifted from C3×D4
ρ182-2-220000-1+√-3-1-√-3001+√-31-√-30000-√2√2-√2√200ζ83ζ3-ζ8ζ3ζ87ζ3-ζ85ζ3ζ83ζ32-ζ8ζ32ζ87ζ32-ζ85ζ32    complex lifted from C3×D8
ρ192-2-220000-1-√-3-1+√-3001-√-31+√-30000√2-√2√2-√200ζ87ζ32-ζ85ζ32ζ83ζ32-ζ8ζ32ζ87ζ3-ζ85ζ3ζ83ζ3-ζ8ζ3    complex lifted from C3×D8
ρ202-2-220000-1-√-3-1+√-3001-√-31+√-30000-√2√2-√2√200ζ83ζ32-ζ8ζ32ζ87ζ32-ζ85ζ32ζ83ζ3-ζ8ζ3ζ87ζ3-ζ85ζ3    complex lifted from C3×D8
ρ212-2-220000-1+√-3-1-√-3001+√-31-√-30000√2-√2√2-√200ζ87ζ3-ζ85ζ3ζ83ζ3-ζ8ζ3ζ87ζ32-ζ85ζ32ζ83ζ32-ζ8ζ32    complex lifted from C3×D8
ρ2233-1-1-3-311003-100000033-1-1000000    orthogonal lifted from C2×A4
ρ2333-1-1-33-11003-1000000-3-311000000    orthogonal lifted from C2×A4
ρ2433-1-133-1-1003-100000033-1-1000000    orthogonal lifted from A4
ρ2533-1-13-31-1003-1000000-3-311000000    orthogonal lifted from C2×A4
ρ2666-2-2000000-620000000000000000    orthogonal lifted from D4×A4
ρ276-62-200000000000000-3√23√2√2-√2000000    orthogonal faithful
ρ286-62-2000000000000003√2-3√2-√2√2000000    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 23)(2 11 24)(3 12 17)(4 13 18)(5 14 19)(6 15 20)(7 16 21)(8 9 22)

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

10000
01000
007200
007201
007210
,
10000
01000
000721
000720
001720
,
10000
01000
00001
00100
00010
,
721000
3933000
00100
00010
00001
,
720000
391000
00100
00010
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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[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
 

Export

Character table of A4×D8 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁