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## G = C6.402+ 1+4order 192 = 26·3

### 40th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.402+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.402+ 1+4
 Lower central C3 — C2×C6 — C6.402+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.402+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 992 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×2], C22 [×27], S3 [×5], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], Dic3 [×2], Dic3 [×4], C12 [×4], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×5], D12 [×2], C2×Dic3 [×5], C2×Dic3 [×4], C3⋊D4 [×4], C3⋊D4 [×7], C2×C12 [×4], C2×C12, C3×D4 [×5], C22×S3 [×3], C22×S3 [×10], C22×C6 [×3], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×3], D6⋊C4 [×7], C6.D4 [×3], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×3], C2×D12, S3×D4 [×4], D42S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×5], C22×C12, C6×D4 [×3], S3×C23 [×2], D45D4, S3×C22⋊C4, Dic34D4, Dic3⋊D4, C23.11D6, D6.D4, D6⋊Q8, C2×D6⋊C4, C4×C3⋊D4, C23.23D6, C232D6 [×2], C23.14D6, C3×C4⋊D4, C2×S3×D4, C2×D42S3, C6.402+ 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, D46D6, S3×C4○D4, C6.402+ 1+4

Smallest permutation representation of C6.402+ 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 28 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 37 22 43)(17 38 23 44)(18 39 24 45)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 24 11 18)(6 23 12 17)(25 43 31 37)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 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), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,37,22,43)(17,38,23,44)(18,39,24,45), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,22,7,16)(2,21,8,15)(3,20,9,14)(4,19,10,13)(5,24,11,18)(6,23,12,17)(25,43,31,37)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,37,28,40)(26,38,29,41)(27,39,30,42)(31,43,34,46)(32,44,35,47)(33,45,36,48)>;`

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

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

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 6G 12A 12B 12C 12D 12E 12F 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 6 12 12 12 12 12 12 size 1 1 1 1 2 2 4 4 6 6 6 6 12 2 2 2 4 4 4 6 6 6 6 12 12 12 2 2 2 4 4 8 8 4 4 4 4 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 D4⋊6D6 S3×C4○D4 kernel C6.402+ 1+4 S3×C22⋊C4 Dic3⋊4D4 Dic3⋊D4 C23.11D6 D6.D4 D6⋊Q8 C2×D6⋊C4 C4×C3⋊D4 C23.23D6 C23⋊2D6 C23.14D6 C3×C4⋊D4 C2×S3×D4 C2×D4⋊2S3 C4⋊D4 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C6 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 4 2 1 1 3 4 1 2 2 2

Matrix representation of C6.402+ 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 1 0 0 0 0 12 0 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 3 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 8 1 0 0 0 0 2 5
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 12 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],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,3,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,12,8] >;`

C6.402+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{40}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.40ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1169);`
`// by ID`

`G=gap.SmallGroup(192,1169);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,136,6278]);`
`// Polycyclic`

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

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