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## G = C8×A4order 96 = 25·3

### Direct product of C8 and A4

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

 Derived series C1 — C22 — C8×A4
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — C8×A4
 Lower central C22 — C8×A4
 Upper central C1 — C8

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 >

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 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 6 6 8 8 8 8 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 3 3 4 4 1 1 3 3 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C8 C12 C24 A4 C2×A4 C4×A4 C8×A4 kernel C8×A4 C4×A4 C22×C8 C2×A4 C22×C4 A4 C23 C22 C8 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 4

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

 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 1 0 0 0 1 1 0 0
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

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