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G = C16.A4order 192 = 26·3

The central extension by C16 of A4

Aliases: C16.A4, Q8.C24, SL2(𝔽3).2C8, D4○C16⋊C3, C8.7(C2×A4), C2.3(C8×A4), C4.5(C4×A4), C8○D4.2C6, C4.A4.3C4, C8.A4.3C2, C4○D4.2C12, SmallGroup(192,204)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C16.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C8○D4 — C8.A4 — C16.A4
 Lower central Q8 — C16.A4
 Upper central C1 — C16

Generators and relations for C16.A4
G = < a,b,c,d | a16=d3=1, b2=c2=a8, ab=ba, ac=ca, ad=da, cbc-1=a8b, dbd-1=a8bc, dcd-1=b >

Smallest permutation representation of C16.A4
On 64 points
Generators in S64
```(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 29 9 21)(2 30 10 22)(3 31 11 23)(4 32 12 24)(5 17 13 25)(6 18 14 26)(7 19 15 27)(8 20 16 28)(33 53 41 61)(34 54 42 62)(35 55 43 63)(36 56 44 64)(37 57 45 49)(38 58 46 50)(39 59 47 51)(40 60 48 52)
(1 45 9 37)(2 46 10 38)(3 47 11 39)(4 48 12 40)(5 33 13 41)(6 34 14 42)(7 35 15 43)(8 36 16 44)(17 61 25 53)(18 62 26 54)(19 63 27 55)(20 64 28 56)(21 49 29 57)(22 50 30 58)(23 51 31 59)(24 52 32 60)
(17 33 53)(18 34 54)(19 35 55)(20 36 56)(21 37 57)(22 38 58)(23 39 59)(24 40 60)(25 41 61)(26 42 62)(27 43 63)(28 44 64)(29 45 49)(30 46 50)(31 47 51)(32 48 52)```

`G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,29,9,21)(2,30,10,22)(3,31,11,23)(4,32,12,24)(5,17,13,25)(6,18,14,26)(7,19,15,27)(8,20,16,28)(33,53,41,61)(34,54,42,62)(35,55,43,63)(36,56,44,64)(37,57,45,49)(38,58,46,50)(39,59,47,51)(40,60,48,52), (1,45,9,37)(2,46,10,38)(3,47,11,39)(4,48,12,40)(5,33,13,41)(6,34,14,42)(7,35,15,43)(8,36,16,44)(17,61,25,53)(18,62,26,54)(19,63,27,55)(20,64,28,56)(21,49,29,57)(22,50,30,58)(23,51,31,59)(24,52,32,60), (17,33,53)(18,34,54)(19,35,55)(20,36,56)(21,37,57)(22,38,58)(23,39,59)(24,40,60)(25,41,61)(26,42,62)(27,43,63)(28,44,64)(29,45,49)(30,46,50)(31,47,51)(32,48,52)>;`

`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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,29,9,21)(2,30,10,22)(3,31,11,23)(4,32,12,24)(5,17,13,25)(6,18,14,26)(7,19,15,27)(8,20,16,28)(33,53,41,61)(34,54,42,62)(35,55,43,63)(36,56,44,64)(37,57,45,49)(38,58,46,50)(39,59,47,51)(40,60,48,52), (1,45,9,37)(2,46,10,38)(3,47,11,39)(4,48,12,40)(5,33,13,41)(6,34,14,42)(7,35,15,43)(8,36,16,44)(17,61,25,53)(18,62,26,54)(19,63,27,55)(20,64,28,56)(21,49,29,57)(22,50,30,58)(23,51,31,59)(24,52,32,60), (17,33,53)(18,34,54)(19,35,55)(20,36,56)(21,37,57)(22,38,58)(23,39,59)(24,40,60)(25,41,61)(26,42,62)(27,43,63)(28,44,64)(29,45,49)(30,46,50)(31,47,51)(32,48,52) );`

`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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,29,9,21),(2,30,10,22),(3,31,11,23),(4,32,12,24),(5,17,13,25),(6,18,14,26),(7,19,15,27),(8,20,16,28),(33,53,41,61),(34,54,42,62),(35,55,43,63),(36,56,44,64),(37,57,45,49),(38,58,46,50),(39,59,47,51),(40,60,48,52)], [(1,45,9,37),(2,46,10,38),(3,47,11,39),(4,48,12,40),(5,33,13,41),(6,34,14,42),(7,35,15,43),(8,36,16,44),(17,61,25,53),(18,62,26,54),(19,63,27,55),(20,64,28,56),(21,49,29,57),(22,50,30,58),(23,51,31,59),(24,52,32,60)], [(17,33,53),(18,34,54),(19,35,55),(20,36,56),(21,37,57),(22,38,58),(23,39,59),(24,40,60),(25,41,61),(26,42,62),(27,43,63),(28,44,64),(29,45,49),(30,46,50),(31,47,51),(32,48,52)]])`

56 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 16A ··· 16H 16I 16J 16K 16L 24A ··· 24H 48A ··· 48P order 1 2 2 3 3 4 4 4 6 6 8 8 8 8 8 8 12 12 12 12 16 ··· 16 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 6 4 4 1 1 6 4 4 1 1 1 1 6 6 4 4 4 4 1 ··· 1 6 6 6 6 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C8 C12 C24 C16.A4 A4 C2×A4 C4×A4 C8×A4 kernel C16.A4 C8.A4 D4○C16 C4.A4 C8○D4 SL2(𝔽3) C4○D4 Q8 C1 C16 C8 C4 C2 # reps 1 1 2 2 2 4 4 8 24 1 1 2 4

Matrix representation of C16.A4 in GL2(𝔽17) generated by

 7 0 0 7
,
 4 10 0 13
,
 13 0 10 4
,
 0 9 15 16
`G:=sub<GL(2,GF(17))| [7,0,0,7],[4,0,10,13],[13,10,0,4],[0,15,9,16] >;`

C16.A4 in GAP, Magma, Sage, TeX

`C_{16}.A_4`
`% in TeX`

`G:=Group("C16.A4");`
`// GroupNames label`

`G:=SmallGroup(192,204);`
`// by ID`

`G=gap.SmallGroup(192,204);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,-2,-2,2,-2,42,58,248,851,172,1524,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=d^3=1,b^2=c^2=a^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^8*b,d*b*d^-1=a^8*b*c,d*c*d^-1=b>;`
`// generators/relations`

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