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## G = D6.C24order 192 = 26·3

### 9th non-split extension by D6 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D6.C24
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4 — D6.C24
 Lower central C3 — C6 — D6.C24
 Upper central C1 — C2 — 2+ 1+4

Generators and relations for D6.C24
G = < a,b,c,d,e,f | a6=b2=c2=e2=f2=1, d2=a3, bab=eae=a-1, ac=ca, ad=da, af=fa, cbc=fbf=a3b, bd=db, ebe=a4b, cd=dc, ce=ec, cf=fc, de=ed, fdf=a3d, fef=a3e >

Subgroups: 1640 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C3, C4 [×6], C4 [×10], C22 [×9], C22 [×21], S3 [×6], C6, C6 [×9], C2×C4 [×9], C2×C4 [×51], D4 [×18], D4 [×42], Q8 [×2], Q8 [×18], C23 [×6], C23 [×9], Dic3, Dic3 [×9], C12 [×6], D6 [×6], D6 [×9], C2×C6 [×9], C2×C6 [×6], C22×C4 [×15], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×15], C4○D4 [×6], C4○D4 [×74], Dic6 [×18], C4×S3 [×24], D12 [×6], C2×Dic3 [×27], C3⋊D4 [×36], C2×C12 [×9], C3×D4 [×18], C3×Q8 [×2], C22×S3 [×9], C22×C6 [×6], C2×C4○D4 [×15], 2+ 1+4, 2+ 1+4 [×9], 2- 1+4 [×6], C2×Dic6 [×9], S3×C2×C4 [×9], C4○D12 [×18], S3×D4 [×18], D42S3 [×54], S3×Q8 [×6], Q83S3 [×2], C22×Dic3 [×6], C2×C3⋊D4 [×18], C6×D4 [×9], C3×C4○D4 [×6], C2.C25, C2×D42S3 [×9], D46D6 [×9], S3×C4○D4 [×6], Q8○D12 [×6], C3×2+ 1+4, D6.C24
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], C25, S3×C23 [×15], C2.C25, S3×C24, D6.C24

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

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

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

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

51 conjugacy classes

 class 1 2A 2B ··· 2J 2K ··· 2P 3 4A ··· 4F 4G 4H 4I ··· 4Q 6A 6B ··· 6J 12A ··· 12F order 1 2 2 ··· 2 2 ··· 2 3 4 ··· 4 4 4 4 ··· 4 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 6 ··· 6 2 2 ··· 2 3 3 6 ··· 6 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 C2.C25 D6.C24 kernel D6.C24 C2×D4⋊2S3 D4⋊6D6 S3×C4○D4 Q8○D12 C3×2+ 1+4 2+ 1+4 C2×D4 C4○D4 C3 C1 # reps 1 9 9 6 6 1 1 9 6 2 1

Matrix representation of D6.C24 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
,
 1 12 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 5 0 0 0 0 5 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 8 0 0 0 0 0 0 5 0 0
,
 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 12 0 0 0 0 0 0 12 0 0
,
 0 12 0 0 0 0 12 0 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
,
 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 12 0 0 0 0 12 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],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,8,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,5,0,0,0,0,0,0,5,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;`

D6.C24 in GAP, Magma, Sage, TeX

`D_6.C_2^4`
`% in TeX`

`G:=Group("D6.C2^4");`
`// GroupNames label`

`G:=SmallGroup(192,1525);`
`// by ID`

`G=gap.SmallGroup(192,1525);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,438,6278]);`
`// 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=e*a*e=a^-1,a*c=c*a,a*d=d*a,a*f=f*a,c*b*c=f*b*f=a^3*b,b*d=d*b,e*b*e=a^4*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^3*d,f*e*f=a^3*e>;`
`// generators/relations`

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