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## G = C24.59D6order 192 = 26·3

### 6th non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.59D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C24 — C24.59D6
 Lower central C3 — C2×C6 — C24.59D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.59D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=cb=bc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 1608 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×12], C3, C4 [×4], C22 [×3], C22 [×8], C22 [×76], S3 [×8], C6, C6 [×6], C6 [×4], C2×C4 [×12], C23, C23 [×6], C23 [×80], Dic3 [×2], C12 [×2], D6 [×8], D6 [×56], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×2], C24, C24 [×18], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×12], C22×S3 [×64], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C2×C22⋊C4 [×5], C25, D6⋊C4 [×8], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×2], C22×C12 [×2], S3×C23 [×6], S3×C23 [×12], C23×C6, C243C4, C2×D6⋊C4 [×4], C2×C6.D4, C6×C22⋊C4, S3×C24, C24.59D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×12], C23, D6 [×3], C22⋊C4 [×12], C22×C4, C2×D4 [×6], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4 [×3], C22≀C2 [×4], D6⋊C4 [×4], S3×C2×C4, C2×D12, S3×D4 [×4], C2×C3⋊D4, C243C4, S3×C22⋊C4 [×2], D6⋊D4 [×2], C2×D6⋊C4, C232D6 [×2], C24.59D6

Smallest permutation representation of C24.59D6
On 48 points
Generators in S48
```(1 23)(2 30)(3 13)(4 32)(5 15)(6 34)(7 17)(8 36)(9 19)(10 26)(11 21)(12 28)(14 43)(16 45)(18 47)(20 37)(22 39)(24 41)(25 48)(27 38)(29 40)(31 42)(33 44)(35 46)
(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)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(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 45 35 22)(2 21 36 44)(3 43 25 20)(4 19 26 42)(5 41 27 18)(6 17 28 40)(7 39 29 16)(8 15 30 38)(9 37 31 14)(10 13 32 48)(11 47 33 24)(12 23 34 46)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 S3×D4 kernel C24.59D6 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 S3×C24 S3×C23 C2×C22⋊C4 C22×S3 C22×C6 C22×C4 C24 C23 C23 C23 C22 # reps 1 4 1 1 1 8 1 8 4 2 1 4 4 4 4

Matrix representation of C24.59D6 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 5 0 0 0 0 0 9 2 0 0 0 11 11 0 0 0 0 0 0 1 0 0 0 12 0
,
 8 0 0 0 0 0 2 9 0 0 0 11 11 0 0 0 0 0 0 1 0 0 0 1 0

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

C24.59D6 in GAP, Magma, Sage, TeX

`C_2^4._{59}D_6`
`% in TeX`

`G:=Group("C2^4.59D6");`
`// GroupNames label`

`G:=SmallGroup(192,514);`
`// by ID`

`G=gap.SmallGroup(192,514);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;`
`// generators/relations`

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