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## G = C6.C25order 192 = 26·3

### 14th non-split extension by C6 of C25 acting via C25/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6.C25
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4 — C6.C25
 Lower central C3 — C6 — C6.C25
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C6.C25
G = < a,b,c,d,e,f | a6=b2=c2=e2=f2=1, d2=a3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, cd=dc, ece=a3c, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 1672 in 810 conjugacy classes, 443 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2.C25, C2×C4○D12, D46D6, Q8.15D6, S3×C4○D4, D4○D12, Q8○D12, C6×C4○D4, C6.C25
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, C25, S3×C23, C2.C25, S3×C24, C6.C25

Smallest permutation representation of C6.C25
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)
(2 6)(3 5)(7 10)(8 9)(11 12)(14 18)(15 17)(19 22)(20 21)(23 24)(26 30)(27 29)(31 34)(32 33)(35 36)(38 42)(39 41)(43 46)(44 45)(47 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)
(1 28 4 25)(2 29 5 26)(3 30 6 27)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(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)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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), (2,6)(3,5)(7,10)(8,9)(11,12)(14,18)(15,17)(19,22)(20,21)(23,24)(26,30)(27,29)(31,34)(32,33)(35,36)(38,42)(39,41)(43,46)(44,45)(47,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,34)(20,35)(21,36)(22,31)(23,32)(24,33), (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (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), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)>;`

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

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

54 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2P 3 4A 4B 4C ··· 4I 4J ··· 4Q 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 ··· 2 2 ··· 2 3 4 4 4 ··· 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 2 ··· 2 6 ··· 6 2 1 1 2 ··· 2 6 ··· 6 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C2.C25 C6.C25 kernel C6.C25 C2×C4○D12 D4⋊6D6 Q8.15D6 S3×C4○D4 D4○D12 Q8○D12 C6×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C3 C1 # reps 1 6 6 2 8 4 4 1 1 3 3 1 8 2 4

Matrix representation of C6.C25 in GL6(𝔽13)

 0 12 0 0 0 0 1 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 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 8 0 0 0 0 0 0 8 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C6.C25 in GAP, Magma, Sage, TeX

`C_6.C_2^5`
`% in TeX`

`G:=Group("C6.C2^5");`
`// GroupNames label`

`G:=SmallGroup(192,1523);`
`// by ID`

`G=gap.SmallGroup(192,1523);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,6278]);`
`// Polycyclic`

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

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