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## G = C24⋊12D6order 192 = 26·3

### 7th semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24⋊12D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C24⋊12D6
 Lower central C3 — C2×C6 — C24⋊12D6
 Upper central C1 — C22 — C22×D4

Generators and relations for C2412D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 968 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×6], C22 [×30], S3 [×2], C6, C6 [×2], C6 [×8], C2×C4 [×2], C2×C4 [×12], D4 [×20], C23 [×3], C23 [×6], C23 [×12], Dic3 [×6], C12 [×2], D6 [×10], C2×C6, C2×C6 [×6], C2×C6 [×20], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×4], C2×D4 [×16], C24 [×2], C24, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×4], C22×C6 [×3], C22×C6 [×6], C22×C6 [×6], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4, C22×D4, Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×8], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23, C23×C6 [×2], C233D4, C23.28D6 [×2], C23.23D6 [×2], C232D6 [×2], C23.14D6 [×4], C2×C6.D4, C244S3 [×2], C22×C3⋊D4, D4×C2×C6, C2412D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×C3⋊D4 [×6], S3×C23, C233D4, D46D6 [×2], C22×C3⋊D4, C2412D6

Smallest permutation representation of C2412D6
On 48 points
Generators in S48
```(1 9)(2 7)(3 8)(4 20)(5 21)(6 19)(10 18)(11 16)(12 17)(13 23)(14 24)(15 22)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)
(1 36)(2 34)(3 32)(4 33)(5 31)(6 35)(7 41)(8 39)(9 37)(10 28)(11 26)(12 30)(13 29)(14 27)(15 25)(16 44)(17 48)(18 46)(19 42)(20 40)(21 38)(22 43)(23 47)(24 45)
(1 15)(2 13)(3 14)(4 10)(5 11)(6 12)(7 23)(8 24)(9 22)(16 21)(17 19)(18 20)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 4)(2 5)(3 6)(7 21)(8 19)(9 20)(10 15)(11 13)(12 14)(16 23)(17 24)(18 22)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 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 3)(4 6)(7 21)(8 20)(9 19)(10 12)(14 15)(16 23)(17 22)(18 24)(25 35)(26 34)(27 33)(28 32)(29 31)(30 36)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C3⋊D4 2+ 1+4 D4⋊6D6 kernel C24⋊12D6 C23.28D6 C23.23D6 C23⋊2D6 C23.14D6 C2×C6.D4 C24⋊4S3 C22×C3⋊D4 D4×C2×C6 C22×D4 C22×C6 C22×C4 C2×D4 C24 C23 C6 C2 # reps 1 2 2 2 4 1 2 1 1 1 4 1 4 2 8 2 4

Matrix representation of C2412D6 in GL6(𝔽13)

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

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

C2412D6 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_{12}D_6`
`% in TeX`

`G:=Group("C2^4:12D6");`
`// GroupNames label`

`G:=SmallGroup(192,1363);`
`// by ID`

`G=gap.SmallGroup(192,1363);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,570,6278]);`
`// Polycyclic`

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

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