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## G = C42.A4order 192 = 26·3

### The non-split extension by C42 of A4 acting faithfully

Aliases: C42.A4, C22.58C24⋊C3, C22.3(C22⋊A4), SmallGroup(192,1025)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22.58C24 — C42.A4
 Chief series C1 — C22 — C42 — C22.58C24 — C42.A4
 Lower central C22.58C24 — C42.A4
 Upper central C1

Generators and relations for C42.A4
G = < a,b,c,d,e | a4=b4=e3=1, c2=b2, d2=a2b2, ab=ba, cac-1=ebe-1=a-1, dad-1=a-1b2, eae-1=a-1b, cbc-1=a2b-1, dbd-1=b-1, cd=dc, ece-1=a2b2cd, ede-1=a2c >

Character table of C42.A4

 class 1 2 3A 3B 4A 4B 4C 4D 4E size 1 3 64 64 12 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ3 ζ32 1 1 1 1 1 linear of order 3 ρ3 1 1 ζ32 ζ3 1 1 1 1 1 linear of order 3 ρ4 3 3 0 0 -1 -1 -1 -1 3 orthogonal lifted from A4 ρ5 3 3 0 0 -1 3 -1 -1 -1 orthogonal lifted from A4 ρ6 3 3 0 0 3 -1 -1 -1 -1 orthogonal lifted from A4 ρ7 3 3 0 0 -1 -1 3 -1 -1 orthogonal lifted from A4 ρ8 3 3 0 0 -1 -1 -1 3 -1 orthogonal lifted from A4 ρ9 12 -4 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C42.A4
On 48 points
Generators in S48
```(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 5 3 6)(2 8 4 7)(9 15 10 16)(11 13 12 14)(17 20 19 18)(21 24 23 22)(25 26 27 28)(33 35)(34 36)(37 38 39 40)(41 43)(42 44)
(1 2 3 4)(5 7 6 8)(9 11 10 12)(13 15 14 16)(17 39 19 37)(18 38 20 40)(21 27 23 25)(22 26 24 28)(29 34)(30 33)(31 36)(32 35)(41 47)(42 46)(43 45)(44 48)
(1 10 3 9)(2 12 4 11)(5 15 6 16)(7 14 8 13)(17 26)(18 27)(19 28)(20 25)(21 40)(22 37)(23 38)(24 39)(29 48 31 46)(30 47 32 45)(33 41 35 43)(34 44 36 42)
(1 46 26)(2 29 39)(3 48 28)(4 31 37)(5 47 25)(6 45 27)(7 32 40)(8 30 38)(9 36 24)(10 34 22)(11 42 19)(12 44 17)(13 41 18)(14 43 20)(15 35 23)(16 33 21)```

`G:=sub<Sym(48)| (9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,5,3,6)(2,8,4,7)(9,15,10,16)(11,13,12,14)(17,20,19,18)(21,24,23,22)(25,26,27,28)(33,35)(34,36)(37,38,39,40)(41,43)(42,44), (1,2,3,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,39,19,37)(18,38,20,40)(21,27,23,25)(22,26,24,28)(29,34)(30,33)(31,36)(32,35)(41,47)(42,46)(43,45)(44,48), (1,10,3,9)(2,12,4,11)(5,15,6,16)(7,14,8,13)(17,26)(18,27)(19,28)(20,25)(21,40)(22,37)(23,38)(24,39)(29,48,31,46)(30,47,32,45)(33,41,35,43)(34,44,36,42), (1,46,26)(2,29,39)(3,48,28)(4,31,37)(5,47,25)(6,45,27)(7,32,40)(8,30,38)(9,36,24)(10,34,22)(11,42,19)(12,44,17)(13,41,18)(14,43,20)(15,35,23)(16,33,21)>;`

`G:=Group( (9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,5,3,6)(2,8,4,7)(9,15,10,16)(11,13,12,14)(17,20,19,18)(21,24,23,22)(25,26,27,28)(33,35)(34,36)(37,38,39,40)(41,43)(42,44), (1,2,3,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,39,19,37)(18,38,20,40)(21,27,23,25)(22,26,24,28)(29,34)(30,33)(31,36)(32,35)(41,47)(42,46)(43,45)(44,48), (1,10,3,9)(2,12,4,11)(5,15,6,16)(7,14,8,13)(17,26)(18,27)(19,28)(20,25)(21,40)(22,37)(23,38)(24,39)(29,48,31,46)(30,47,32,45)(33,41,35,43)(34,44,36,42), (1,46,26)(2,29,39)(3,48,28)(4,31,37)(5,47,25)(6,45,27)(7,32,40)(8,30,38)(9,36,24)(10,34,22)(11,42,19)(12,44,17)(13,41,18)(14,43,20)(15,35,23)(16,33,21) );`

`G=PermutationGroup([[(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,5,3,6),(2,8,4,7),(9,15,10,16),(11,13,12,14),(17,20,19,18),(21,24,23,22),(25,26,27,28),(33,35),(34,36),(37,38,39,40),(41,43),(42,44)], [(1,2,3,4),(5,7,6,8),(9,11,10,12),(13,15,14,16),(17,39,19,37),(18,38,20,40),(21,27,23,25),(22,26,24,28),(29,34),(30,33),(31,36),(32,35),(41,47),(42,46),(43,45),(44,48)], [(1,10,3,9),(2,12,4,11),(5,15,6,16),(7,14,8,13),(17,26),(18,27),(19,28),(20,25),(21,40),(22,37),(23,38),(24,39),(29,48,31,46),(30,47,32,45),(33,41,35,43),(34,44,36,42)], [(1,46,26),(2,29,39),(3,48,28),(4,31,37),(5,47,25),(6,45,27),(7,32,40),(8,30,38),(9,36,24),(10,34,22),(11,42,19),(12,44,17),(13,41,18),(14,43,20),(15,35,23),(16,33,21)]])`

Matrix representation of C42.A4 in GL12(𝔽13)

 1 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 5 0 0 0
,
 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

`G:=sub<GL(12,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0],[0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C42.A4 in GAP, Magma, Sage, TeX

`C_4^2.A_4`
`% in TeX`

`G:=Group("C4^2.A4");`
`// GroupNames label`

`G:=SmallGroup(192,1025);`
`// by ID`

`G=gap.SmallGroup(192,1025);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,680,2207,184,675,570,745,360,4624,1971,718,102,4037,7062]);`
`// Polycyclic`

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

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