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## G = C12.8C42order 192 = 26·3

### 1st non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.8C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C12 — C23.26D6 — C12.8C42
 Lower central C3 — C6 — C12 — C12.8C42
 Upper central C1 — C2×C4 — C22×C4 — C2×C42

Generators and relations for C12.8C42
G = < a,b,c | a12=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a3b >

Subgroups: 232 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C4×C12, C22×C12, C22×C12, C426C4, C2×C4.Dic3, C23.26D6, C2×C4×C12, C12.8C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4≀C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C426C4, C424S3, C6.C42, C12.8C42

Smallest permutation representation of C12.8C42
On 48 points
Generators in S48
```(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)
(1 37 23 25)(2 48 24 36)(3 47 13 35)(4 46 14 34)(5 45 15 33)(6 44 16 32)(7 43 17 31)(8 42 18 30)(9 41 19 29)(10 40 20 28)(11 39 21 27)(12 38 22 26)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 6A ··· 6G 8A 8B 8C 8D 12A ··· 12X order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 2 1 1 1 1 2 ··· 2 12 12 12 12 2 ··· 2 12 12 12 12 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + - + image C1 C2 C2 C2 C4 C4 C4 S3 D4 Q8 D4 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 C3⋊D4 C4≀C2 C42⋊4S3 kernel C12.8C42 C2×C4.Dic3 C23.26D6 C2×C4×C12 C4.Dic3 C4⋊Dic3 C4×C12 C2×C42 C2×C12 C2×C12 C22×C6 C42 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 1 1 1 4 4 4 1 2 1 1 2 1 2 4 2 2 2 8 16

Matrix representation of C12.8C42 in GL3(𝔽73) generated by

 1 0 0 0 24 0 0 0 70
,
 46 0 0 0 0 1 0 1 0
,
 27 0 0 0 72 0 0 0 27
`G:=sub<GL(3,GF(73))| [1,0,0,0,24,0,0,0,70],[46,0,0,0,0,1,0,1,0],[27,0,0,0,72,0,0,0,27] >;`

C12.8C42 in GAP, Magma, Sage, TeX

`C_{12}._8C_4^2`
`% in TeX`

`G:=Group("C12.8C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,82);`
`// by ID`

`G=gap.SmallGroup(192,82);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,1123,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;`
`// generators/relations`

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