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## G = C6.1222+ 1+4order 192 = 26·3

### 31st 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 — C6.1222+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4 — C6.1222+ 1+4
 Lower central C3 — C2×C6 — C6.1222+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.1222+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=eae-1=a-1, ac=ca, ad=da, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 672 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C22.45C24, Dic3.D4, S3×C22⋊C4, Dic34D4, C23.9D6, C23.11D6, C23.21D6, C4⋊C47S3, C4.D12, C4⋊C4⋊S3, C23.26D6, C2×D6⋊C4, D4×Dic3, C232D6, C3×C22.D4, C6.1222+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D42S3, S3×C23, C22.45C24, C2×D42S3, S3×C4○D4, D4○D12, C6.1222+ 1+4

Smallest permutation representation of C6.1222+ 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 43 18 9)(2 48 13 8)(3 47 14 7)(4 46 15 12)(5 45 16 11)(6 44 17 10)(19 41 29 31)(20 40 30 36)(21 39 25 35)(22 38 26 34)(23 37 27 33)(24 42 28 32)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 38)(20 39)(21 40)(22 41)(23 42)(24 37)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 22 18 26)(2 23 13 27)(3 24 14 28)(4 19 15 29)(5 20 16 30)(6 21 17 25)(7 35 47 39)(8 36 48 40)(9 31 43 41)(10 32 44 42)(11 33 45 37)(12 34 46 38)
(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)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 2 2 3 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 12 2 2 2 4 4 4 4 6 ··· 6 12 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 D6 D6 D6 D6 C4○D4 C4○D4 2+ 1+4 D4⋊2S3 S3×C4○D4 D4○D12 kernel C6.1222+ 1+4 Dic3.D4 S3×C22⋊C4 Dic3⋊4D4 C23.9D6 C23.11D6 C23.21D6 C4⋊C4⋊7S3 C4.D12 C4⋊C4⋊S3 C23.26D6 C2×D6⋊C4 D4×Dic3 C23⋊2D6 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C2×C6 C6 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 3 2 1 1 4 4 1 2 2 2

Matrix representation of C6.1222+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 4 9 0 0 0 0 1 9
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 9 0 0 0 0 7 9
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 3 5
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,4,1,0,0,0,0,9,9],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,7,0,0,0,0,9,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,3,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;`

C6.1222+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{122}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.122ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1217);`
`// by ID`

`G=gap.SmallGroup(192,1217);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,136,6278]);`
`// Polycyclic`

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

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