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G = C8×A4order 96 = 25·3

Direct product of C8 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C8×A4, C22⋊C24, C23.2C12, (C22×C8)⋊C3, C4.4(C2×A4), C2.1(C4×A4), (C4×A4).4C2, (C2×A4).2C4, (C22×C4).2C6, SmallGroup(96,73)

Series: Derived Chief Lower central Upper central

C1C22 — C8×A4
C1C22C23C22×C4C4×A4 — C8×A4
C22 — C8×A4
C1C8

Generators and relations for C8×A4
 G = < a,b,c,d | a8=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
3C2
4C3
3C22
3C4
3C22
4C6
3C2×C4
3C8
3C2×C4
4C12
3C2×C8
3C2×C8
4C24

Permutation representations of C8×A4
On 24 points - transitive group 24T85
Generators in S24
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 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 15 19)(2 16 20)(3 9 21)(4 10 22)(5 11 23)(6 12 24)(7 13 17)(8 14 18)

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

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

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

G:=TransitiveGroup(24,85);

C8×A4 is a maximal subgroup of   A4⋊C16  A4⋊Q16  C8⋊S4  C82S4  A4⋊D8
C8×A4 is a maximal quotient of   C16.A4

32 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B8A8B8C8D8E8F8G8H12A12B12C12D24A···24H
order122233444466888888881212121224···24
size1133441133441111333344444···4

32 irreducible representations

dim111111113333
type++++
imageC1C2C3C4C6C8C12C24A4C2×A4C4×A4C8×A4
kernelC8×A4C4×A4C22×C8C2×A4C22×C4A4C23C22C8C4C2C1
# reps112224481124

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

2200
0220
0022
,
100
0720
0072
,
7200
0720
001
,
010
001
100
G:=sub<GL(3,GF(73))| [22,0,0,0,22,0,0,0,22],[1,0,0,0,72,0,0,0,72],[72,0,0,0,72,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;

C8×A4 in GAP, Magma, Sage, TeX

C_8\times A_4
% in TeX

G:=Group("C8xA4");
// GroupNames label

G:=SmallGroup(96,73);
// by ID

G=gap.SmallGroup(96,73);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,2,36,50,730,1307]);
// Polycyclic

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

Export

Subgroup lattice of C8×A4 in TeX

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