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## G = A4×M4(2)  order 192 = 26·3

### Direct product of A4 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — A4×M4(2)
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — C2×C4×A4 — A4×M4(2)
 Lower central C22 — C23 — A4×M4(2)
 Upper central C1 — C4 — M4(2)

Generators and relations for A4×M4(2)
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=d5 >

Subgroups: 224 in 83 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, C23, C23, C12, A4, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C24, C2×C12, C2×A4, C2×A4, C22×C8, C2×M4(2), C23×C4, C3×M4(2), C4×A4, C22×A4, C22×M4(2), C8×A4, C2×C4×A4, A4×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, M4(2), C2×C12, C2×A4, C3×M4(2), C4×A4, C22×A4, C2×C4×A4, A4×M4(2)

Permutation representations of A4×M4(2)
On 24 points - transitive group 24T297
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 6)(4 8)(9 13)(11 15)(18 22)(20 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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)>;

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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24) );

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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24)]])

G:=TransitiveGroup(24,297);

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 6 type + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) A4 C2×A4 C2×A4 C4×A4 C4×A4 A4×M4(2) kernel A4×M4(2) C8×A4 C2×C4×A4 C22×M4(2) C4×A4 C22×A4 C22×C8 C23×C4 C22×C4 C24 A4 C22 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 2 1 2 2 2

Matrix representation of A4×M4(2) in GL5(𝔽73)

 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 0 1 0 0 1 0 0 0 0 0 1 0
,
 0 46 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 46
,
 72 0 0 0 0 0 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))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,0,1,0,0,72,1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×M4(2) in GAP, Magma, Sage, TeX

A_4\times M_4(2)
% in TeX

G:=Group("A4xM4(2)");
// GroupNames label

G:=SmallGroup(192,1011);
// by ID

G=gap.SmallGroup(192,1011);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,365,92,80,1027,1784]);
// 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^5>;
// generators/relations

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