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## G = C6.612+ 1+4order 192 = 26·3

### 61st 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 — C6.612+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C6.612+ 1+4
 Lower central C3 — C2×C6 — C6.612+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.612+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a3b-1, dbd-1=ebe=a3b, dcd-1=ece=a3c, ede=a3b2d >

Subgroups: 752 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], Dic3 [×5], C12 [×5], D6 [×2], D6 [×12], C2×C6, C2×C6 [×6], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic6, C4×S3 [×3], D12 [×3], C2×Dic3 [×5], C3⋊D4 [×5], C2×C12 [×5], C2×C12, C3×D4, C22×S3 [×3], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4 [×2], C422C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×3], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×3], C2×D12 [×2], C2×C3⋊D4 [×4], C22×C12, C6×D4, S3×C23, C22.32C24, C23.8D6, S3×C22⋊C4, D6⋊D4, Dic3⋊D4, C23.11D6 [×2], Dic35D4, D6.D4, C4.D12, C4⋊C4⋊S3, C4×C3⋊D4, C127D4, C232D6, D63D4, C3×C22.D4, C6.612+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ 1+4 [×2], S3×C23, C22.32C24, D46D6, S3×C4○D4, D4○D12, C6.612+ 1+4

Smallest permutation representation of C6.612+ 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 25 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 19 7 13)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,43,31,37)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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), (1,25,10,34)(2,26,11,35)(3,27,12,36)(4,28,7,31)(5,29,8,32)(6,30,9,33)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,43,31,37)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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)], [(1,25,10,34),(2,26,11,35),(3,27,12,36),(4,28,7,31),(5,29,8,32),(6,30,9,33),(13,40,22,43),(14,41,23,44),(15,42,24,45),(16,37,19,46),(17,38,20,47),(18,39,21,48)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,19,7,13),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14),(25,43,31,37),(26,48,32,42),(27,47,33,41),(28,46,34,40),(29,45,35,39),(30,44,36,38)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)])`

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 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 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 4 4 6 6 12 12 2 2 2 4 4 4 4 6 6 12 12 12 12 2 2 2 4 4 8 4 4 4 4 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 2+ 1+4 D4⋊6D6 S3×C4○D4 D4○D12 kernel C6.612+ 1+4 C23.8D6 S3×C22⋊C4 D6⋊D4 Dic3⋊D4 C23.11D6 Dic3⋊5D4 D6.D4 C4.D12 C4⋊C4⋊S3 C4×C3⋊D4 C12⋊7D4 C23⋊2D6 D6⋊3D4 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 C6 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 2 1 1 4 2 2 2 2

Matrix representation of C6.612+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 5 0 0 0 0 0 6 8 0 0 0 0 0 0 4 6 10 3 0 0 7 10 10 0 0 0 11 0 9 7 0 0 0 11 6 3
,
 12 0 0 0 0 0 4 1 0 0 0 0 0 0 1 0 4 6 0 0 0 1 7 10 0 0 0 0 12 0 0 0 0 0 0 12
,
 7 10 0 0 0 0 3 6 0 0 0 0 0 0 11 11 5 8 0 0 9 2 0 8 0 0 0 0 11 11 0 0 0 0 9 2
,
 6 3 0 0 0 0 10 7 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0 4 11

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,6,0,0,0,0,0,8,0,0,0,0,0,0,4,7,11,0,0,0,6,10,0,11,0,0,10,10,9,6,0,0,3,0,7,3],[12,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,4,7,12,0,0,0,6,10,0,12],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,11,9,0,0,0,0,11,2,0,0,0,0,5,0,11,9,0,0,8,8,11,2],[6,10,0,0,0,0,3,7,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;`

C6.612+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{61}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.61ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1216);`
`// by ID`

`G=gap.SmallGroup(192,1216);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,6278]);`
`// Polycyclic`

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

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