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## G = M4(2).A4order 192 = 26·3

### The non-split extension by M4(2) of A4 acting through Inn(M4(2))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — M4(2).A4
 Chief series C1 — C2 — Q8 — C4○D4 — C4.A4 — C2×C4.A4 — M4(2).A4
 Lower central Q8 — M4(2).A4
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2).A4
G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Subgroups: 197 in 73 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C2×C12, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C3×M4(2), C2×SL2(𝔽3), C4.A4, Q8○M4(2), C8.A4, C2×C4.A4, M4(2).A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C2×C4×A4, M4(2).A4

Smallest permutation representation of M4(2).A4
On 32 points
Generators in S32
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)
(1 16 5 12)(2 9 6 13)(3 10 7 14)(4 11 8 15)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(9 29 20)(10 30 21)(11 31 22)(12 32 23)(13 25 24)(14 26 17)(15 27 18)(16 28 19)```

`G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (9,29,20)(10,30,21)(11,31,22)(12,32,23)(13,25,24)(14,26,17)(15,27,18)(16,28,19)>;`

`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)(25,26,27,28,29,30,31,32), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (9,29,20)(10,30,21)(11,31,22)(12,32,23)(13,25,24)(14,26,17)(15,27,18)(16,28,19) );`

`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),(25,26,27,28,29,30,31,32)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28)], [(1,16,5,12),(2,9,6,13),(3,10,7,14),(4,11,8,15),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29)], [(9,29,20),(10,30,21),(11,31,22),(12,32,23),(13,25,24),(14,26,17),(15,27,18),(16,28,19)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 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 3 3 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 6 6 4 4 1 1 2 6 6 4 4 8 8 2 2 2 2 6 6 6 6 4 4 4 4 8 8 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 4 type + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 A4 C2×A4 C2×A4 C4×A4 C4×A4 M4(2).A4 kernel M4(2).A4 C8.A4 C2×C4.A4 Q8○M4(2) C2×SL2(𝔽3) C4.A4 C8○D4 C2×C4○D4 C2×Q8 C4○D4 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 2 1 2 2 6

Matrix representation of M4(2).A4 in GL4(𝔽5) generated by

 0 1 0 0 2 0 0 0 0 0 0 4 0 0 3 0
,
 4 0 0 0 0 1 0 0 0 0 4 0 0 0 0 1
,
 3 0 0 0 0 3 0 0 2 0 2 0 0 3 0 2
,
 2 0 4 0 0 2 0 1 0 0 3 0 0 0 0 3
,
 4 0 2 0 0 4 0 3 2 0 0 0 0 3 0 0
`G:=sub<GL(4,GF(5))| [0,2,0,0,1,0,0,0,0,0,0,3,0,0,4,0],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[3,0,2,0,0,3,0,3,0,0,2,0,0,0,0,2],[2,0,0,0,0,2,0,0,4,0,3,0,0,1,0,3],[4,0,2,0,0,4,0,3,2,0,0,0,0,3,0,0] >;`

M4(2).A4 in GAP, Magma, Sage, TeX

`M_4(2).A_4`
`% in TeX`

`G:=Group("M4(2).A4");`
`// GroupNames label`

`G:=SmallGroup(192,1013);`
`// by ID`

`G=gap.SmallGroup(192,1013);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,1373,92,248,438,172,775,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=b^2=e^3=1,c^2=d^2=a^4,b*a*b=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^4*c,e*c*e^-1=a^4*c*d,e*d*e^-1=c>;`
`// generators/relations`

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