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## G = C24.35D6order 192 = 26·3

### 24th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C24.35D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C24.35D6
 Lower central C3 — C6 — C24.35D6
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C24.35D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 840 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×12], C22, C22 [×6], C22 [×22], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×16], C23 [×3], C23 [×4], C23 [×10], Dic3 [×4], Dic3 [×4], C12 [×4], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C2×C6 [×10], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×12], C24, C24, C4×S3 [×4], C2×Dic3 [×10], C2×Dic3 [×2], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×2], C22×S3 [×6], C22×S3 [×2], C22×C6 [×3], C22×C6 [×4], C22×C6 [×2], C2×C22⋊C4, C2×C22⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×12], C22×C12 [×2], S3×C23, C23×C6, C22.11C24, C23.16D6 [×2], S3×C22⋊C4 [×2], Dic34D4 [×4], C4×C3⋊D4 [×4], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.35D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4 [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D46D6 [×2], C24.35D6

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 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 C4 S3 D6 D6 D6 C4×S3 2+ 1+4 D4⋊6D6 kernel C24.35D6 C23.16D6 S3×C22⋊C4 Dic3⋊4D4 C4×C3⋊D4 C2×C6.D4 C6×C22⋊C4 C22×C3⋊D4 C2×C3⋊D4 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C6 C2 # reps 1 2 2 4 4 1 1 1 16 1 4 2 1 8 2 4

Matrix representation of C24.35D6 in GL6(𝔽13)

 12 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 8 0 12 0 0 0 0 8 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 9 0 0 0 0 4 2 0 0 0 0 0 0 11 9 0 0 0 0 4 2
,
 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 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 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
,
 8 8 0 0 0 0 5 0 0 0 0 0 0 0 5 5 2 2 0 0 8 0 11 0 0 0 0 0 8 8 0 0 0 0 5 0
,
 5 5 0 0 0 0 0 8 0 0 0 0 0 0 5 5 2 2 0 0 0 8 0 11 0 0 0 0 8 8 0 0 0 0 0 5

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

C24.35D6 in GAP, Magma, Sage, TeX

`C_2^4._{35}D_6`
`% in TeX`

`G:=Group("C2^4.35D6");`
`// GroupNames label`

`G:=SmallGroup(192,1045);`
`// by ID`

`G=gap.SmallGroup(192,1045);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,6278]);`
`// Polycyclic`

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

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