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G = C24.12D6order 192 = 26·3

1st non-split extension by C24 of 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×C6 — C23×C6 — C2×C6.D4 — C24.12D6
 Lower central C3 — C6 — C2×C6 — C24.12D6
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.12D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=abc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde5 >

Subgroups: 408 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.9D4, C2×C6.D4, C6×C22⋊C4, C24.12D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C23⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C23.9D4, C6.C42, C23.7D6, C24.12D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + - + + image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 C23⋊C4 C23.7D6 kernel C24.12D6 C2×C6.D4 C6×C22⋊C4 C6.D4 C22×C12 C2×C22⋊C4 C22×C6 C22×C6 C22×C4 C24 C23 C23 C23 C23 C6 C2 # reps 1 2 1 8 4 1 3 1 2 1 2 4 2 4 2 4

Matrix representation of C24.12D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 5 5 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 5 12 11 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 11 2 0 0 0 0 0 0 11 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 5 5 12 11 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 8
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 5 5 12 11 0 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 8 8

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,11,0,8],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,5,0,1,12,0,0,0,0,5,0,0,0,0,0,0,0,12,12,0,8,0,0,0,0,11,0,0,8] >;`

C24.12D6 in GAP, Magma, Sage, TeX

`C_2^4._{12}D_6`
`% in TeX`

`G:=Group("C2^4.12D6");`
`// GroupNames label`

`G:=SmallGroup(192,85);`
`// by ID`

`G=gap.SmallGroup(192,85);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1684,6278]);`
`// Polycyclic`

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

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