Copied to
clipboard

## G = C6.1212+ 1+4order 192 = 26·3

### 30th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.1212+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4 — C6.1212+ 1+4
 Lower central C3 — C2×C6 — C6.1212+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.1212+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1024 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×2], C22 [×27], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×2], Dic3 [×3], C12 [×5], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×7], D12 [×7], C2×Dic3 [×4], C2×Dic3, C3⋊D4 [×4], C3⋊D4 [×5], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×4], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×8], C6.D4, C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×4], C2×D12 [×4], C4○D12 [×4], S3×D4 [×4], C22×Dic3, C2×C3⋊D4 [×4], C22×C12, C6×D4, S3×C23 [×2], D45D4, S3×C22⋊C4, Dic34D4, D6⋊D4 [×2], Dic3⋊D4, C23.11D6, Dic35D4, D6.D4, C12⋊D4, D6⋊Q8, C2×D6⋊C4, C23.14D6, C3×C22.D4, C2×C4○D12, C2×S3×D4, C6.1212+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], S3×C23, D45D4, C2×S3×D4, S3×C4○D4, D4○D12, C6.1212+ 1+4

Smallest permutation representation of C6.1212+ 1+4
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 29 15 22)(2 30 16 23)(3 25 17 24)(4 26 18 19)(5 27 13 20)(6 28 14 21)(7 35 44 42)(8 36 45 37)(9 31 46 38)(10 32 47 39)(11 33 48 40)(12 34 43 41)
(1 29 4 26)(2 28 5 25)(3 27 6 30)(7 37 10 40)(8 42 11 39)(9 41 12 38)(13 24 16 21)(14 23 17 20)(15 22 18 19)(31 46 34 43)(32 45 35 48)(33 44 36 47)
(1 31 15 38)(2 36 16 37)(3 35 17 42)(4 34 18 41)(5 33 13 40)(6 32 14 39)(7 25 44 24)(8 30 45 23)(9 29 46 22)(10 28 47 21)(11 27 48 20)(12 26 43 19)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 35)(14 36)(15 31)(16 32)(17 33)(18 34)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 2 2 4 6 6 6 6 12 12 2 2 2 4 4 4 4 6 6 6 6 12 12 2 2 2 4 4 8 4 4 4 4 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 S3×C4○D4 D4○D12 kernel C6.1212+ 1+4 S3×C22⋊C4 Dic3⋊4D4 D6⋊D4 Dic3⋊D4 C23.11D6 Dic3⋊5D4 D6.D4 C12⋊D4 D6⋊Q8 C2×D6⋊C4 C23.14D6 C3×C22.D4 C2×C4○D12 C2×S3×D4 C22.D4 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C6 C22 C2 C2 # reps 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 4 3 2 1 1 4 1 2 2 2

Matrix representation of C6.1212+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 5 0 0 0 0 10 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 5 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;`

C6.1212+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{121}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.121ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1213);`
`// by ID`

`G=gap.SmallGroup(192,1213);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;`
`// generators/relations`

׿
×
𝔽