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## G = C24.53D6order 192 = 26·3

### 42nd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.53D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×D4⋊2S3 — C24.53D6
 Lower central C3 — C2×C6 — C24.53D6
 Upper central C1 — C22 — C22×D4

Generators and relations for C24.53D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 824 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], S3, C6 [×3], C6 [×8], C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×11], Dic3 [×7], C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×6], C2×C6 [×20], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×4], C22×C6 [×10], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4, C6.D4, C6.D4 [×10], C2×Dic6, S3×C2×C4, D42S3 [×4], C22×Dic3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4 [×2], C6×D4 [×2], C6×D4 [×4], C23×C6 [×2], D45D4, C12.48D4, C4×C3⋊D4, D4×Dic3, C23.23D6 [×2], C23.12D6, D63D4, C23.14D6 [×2], C2×C6.D4 [×2], C244S3 [×2], C2×D42S3, D4×C2×C6, C24.53D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, D42S3 [×2], C2×C3⋊D4 [×6], S3×C23, D45D4, C2×D42S3, D46D6, C22×C3⋊D4, C24.53D6

Smallest permutation representation of C24.53D6
On 48 points
Generators in S48
```(1 21)(2 16)(3 23)(4 18)(5 13)(6 20)(7 15)(8 22)(9 17)(10 24)(11 19)(12 14)(25 42)(26 37)(27 44)(28 39)(29 46)(30 41)(31 48)(32 43)(33 38)(34 45)(35 40)(36 47)
(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 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)
(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 47 7 41)(2 40 8 46)(3 45 9 39)(4 38 10 44)(5 43 11 37)(6 48 12 42)(13 32 19 26)(14 25 20 31)(15 30 21 36)(16 35 22 29)(17 28 23 34)(18 33 24 27)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 2+ 1+4 D4⋊2S3 D4⋊6D6 kernel C24.53D6 C12.48D4 C4×C3⋊D4 D4×Dic3 C23.23D6 C23.12D6 D6⋊3D4 C23.14D6 C2×C6.D4 C24⋊4S3 C2×D4⋊2S3 D4×C2×C6 C22×D4 C3×D4 C22×C4 C2×D4 C24 C2×C6 D4 C6 C22 C2 # reps 1 1 1 1 2 1 1 2 2 2 1 1 1 4 1 4 2 4 8 1 2 2

Matrix representation of C24.53D6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 11 12
,
 1 0 0 0 0 12 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
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 10 0 0 0 0 4 0 0 0 0 1 1 0 0 11 12
,
 0 9 0 0 3 0 0 0 0 0 8 0 0 0 0 8
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,4,0,0,0,0,1,11,0,0,1,12],[0,3,0,0,9,0,0,0,0,0,8,0,0,0,0,8] >;`

C24.53D6 in GAP, Magma, Sage, TeX

`C_2^4._{53}D_6`
`% in TeX`

`G:=Group("C2^4.53D6");`
`// GroupNames label`

`G:=SmallGroup(192,1365);`
`// by ID`

`G=gap.SmallGroup(192,1365);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,675,6278]);`
`// 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,e*a*e^-1=a*d=d*a,a*f=f*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`

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