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G = A4⋊C8order 96 = 25·3

The semidirect product of A4 and C8 acting via C8/C4=C2

non-abelian, soluble, monomial

Aliases: A4⋊C8, C4.4S4, C23.Dic3, (C2×A4).C4, C22⋊(C3⋊C8), (C4×A4).2C2, C2.1(A4⋊C4), (C22×C4).1S3, SmallGroup(96,65)

Series: Derived Chief Lower central Upper central

C1C22A4 — A4⋊C8
C1C22A4C2×A4C4×A4 — A4⋊C8
A4 — A4⋊C8
C1C4

Generators and relations for A4⋊C8
 G = < a,b,c,d | a2=b2=c3=d8=1, cac-1=dad-1=ab=ba, cbc-1=a, bd=db, dcd-1=c-1 >

3C2
3C2
4C3
3C22
3C4
3C22
4C6
3C2×C4
3C2×C4
6C8
6C8
4C12
3C2×C8
3C2×C8
4C3⋊C8
3C22⋊C8

Character table of A4⋊C8

 class 12A2B2C34A4B4C4D68A8B8C8D8E8F8G8H12A12B
 size 11338113386666666688
ρ111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311111-1-1-1-11-iiii-i-i-ii-1-1    linear of order 4
ρ411111-1-1-1-11i-i-i-iiii-i-1-1    linear of order 4
ρ51-11-11-iii-i-1ζ85ζ83ζ87ζ87ζ8ζ8ζ85ζ83-ii    linear of order 8
ρ61-11-11-iii-i-1ζ8ζ87ζ83ζ83ζ85ζ85ζ8ζ87-ii    linear of order 8
ρ71-11-11i-i-ii-1ζ87ζ8ζ85ζ85ζ83ζ83ζ87ζ8i-i    linear of order 8
ρ81-11-11i-i-ii-1ζ83ζ85ζ8ζ8ζ87ζ87ζ83ζ85i-i    linear of order 8
ρ92222-12222-100000000-1-1    orthogonal lifted from S3
ρ102222-1-2-2-2-2-10000000011    symplectic lifted from Dic3, Schur index 2
ρ112-22-2-1-2i2i2i-2i100000000i-i    complex lifted from C3⋊C8
ρ122-22-2-12i-2i-2i2i100000000-ii    complex lifted from C3⋊C8
ρ1333-1-1033-1-1011-11-11-1-100    orthogonal lifted from S4
ρ1433-1-1033-1-10-1-11-11-11100    orthogonal lifted from S4
ρ1533-1-10-3-3110-ii-iii-ii-i00    complex lifted from A4⋊C4
ρ1633-1-10-3-3110i-ii-i-ii-ii00    complex lifted from A4⋊C4
ρ173-3-110-3i3i-ii0ζ8ζ87ζ87ζ83ζ8ζ85ζ85ζ8300    complex faithful
ρ183-3-110-3i3i-ii0ζ85ζ83ζ83ζ87ζ85ζ8ζ8ζ8700    complex faithful
ρ193-3-1103i-3ii-i0ζ87ζ8ζ8ζ85ζ87ζ83ζ83ζ8500    complex faithful
ρ203-3-1103i-3ii-i0ζ83ζ85ζ85ζ8ζ83ζ87ζ87ζ800    complex faithful

Permutation representations of A4⋊C8
On 24 points - transitive group 24T89
Generators in S24
(1 5)(3 7)(9 13)(10 14)(11 15)(12 16)(18 22)(20 24)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 19 11)(2 12 20)(3 21 13)(4 14 22)(5 23 15)(6 16 24)(7 17 9)(8 10 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)

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

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

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

G:=TransitiveGroup(24,89);

A4⋊C8 is a maximal subgroup of
C8×S4  C8⋊S4  A4⋊M4(2)  A4⋊SD16  D4⋊S4  A42Q16  Q83S4  C12.12S4  A5⋊C8  C20.S4  Dic5.S4
A4⋊C8 is a maximal quotient of
C2.U2(𝔽3)  A4⋊C16  C8.7S4  C12.S4  C12.12S4  C20.S4  Dic5.S4

Matrix representation of A4⋊C8 in GL3(𝔽73) generated by

7200
0720
7201
,
7200
7210
0072
,
1710
0721
0720
,
63020
06310
0010
G:=sub<GL(3,GF(73))| [72,0,72,0,72,0,0,0,1],[72,72,0,0,1,0,0,0,72],[1,0,0,71,72,72,0,1,0],[63,0,0,0,63,0,20,10,10] >;

A4⋊C8 in GAP, Magma, Sage, TeX

A_4\rtimes C_8
% in TeX

G:=Group("A4:C8");
// GroupNames label

G:=SmallGroup(96,65);
// by ID

G=gap.SmallGroup(96,65);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,12,31,387,1444,202,869,347]);
// Polycyclic

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

Export

Subgroup lattice of A4⋊C8 in TeX
Character table of A4⋊C8 in TeX

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