Copied to
clipboard

## G = C24.83D6order 192 = 26·3

### 12nd non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.83D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C24.83D6
 Lower central C3 — C2×C6 — C24.83D6
 Upper central C1 — C2×C4 — C23×C4

Generators and relations for C24.83D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 760 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], S3 [×2], C6, C6 [×2], C6 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], Dic3 [×6], C12 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C2×C6 [×14], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×6], C2×C12 [×10], C22×S3 [×2], C22×C6, C22×C6 [×2], C22×C6 [×6], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×6], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×6], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], C23×C6, C22.19C24, C12.48D4 [×2], C23.26D6, C4×C3⋊D4 [×4], C23.28D6 [×2], C127D4 [×2], C244S3 [×2], C2×C4○D12, C23×C12, C24.83D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C4○D12 [×4], C2×C3⋊D4 [×6], S3×C23, C22.19C24, C2×C4○D12 [×2], C22×C3⋊D4, C24.83D6

Smallest permutation representation of C24.83D6
On 48 points
Generators in S48
```(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 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 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 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 41 19 47)(14 46 20 40)(15 39 21 45)(16 44 22 38)(17 37 23 43)(18 42 24 48)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 3 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 6A ··· 6O 12A ··· 12P order 1 2 2 2 2 ··· 2 2 2 3 4 4 4 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 12 12 2 1 1 1 1 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C3⋊D4 C4○D12 kernel C24.83D6 C12.48D4 C23.26D6 C4×C3⋊D4 C23.28D6 C12⋊7D4 C24⋊4S3 C2×C4○D12 C23×C12 C23×C4 C2×C12 C22×C4 C24 C2×C6 C2×C4 C22 # reps 1 2 1 4 2 2 2 1 1 1 4 6 1 8 8 16

Matrix representation of C24.83D6 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 7 0 0 0 0 11 0 0 0 0 9 0 0 0 0 3
,
 0 11 0 0 7 0 0 0 0 0 0 3 0 0 9 0
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,11,0,0,0,0,9,0,0,0,0,3],[0,7,0,0,11,0,0,0,0,0,0,9,0,0,3,0] >;`

C24.83D6 in GAP, Magma, Sage, TeX

`C_2^4._{83}D_6`
`% in TeX`

`G:=Group("C2^4.83D6");`
`// GroupNames label`

`G:=SmallGroup(192,1350);`
`// by ID`

`G=gap.SmallGroup(192,1350);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,6278]);`
`// Polycyclic`

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

׿
×
𝔽