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## G = C24.52D6order 192 = 26·3

### 41st non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.52D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C24.52D6
 Lower central C3 — C2×C6 — C24.52D6
 Upper central C1 — C22 — C22×D4

Generators and relations for C24.52D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 904 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C22×D4, C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C23×C6, C22.29C24, C23.26D6, C23.12D6, D63D4, C123D4, C244S3, C2×C4○D12, D4×C2×C6, C24.52D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, 2+ 1+4, C2×C3⋊D4, S3×C23, C22.29C24, D46D6, C22×C3⋊D4, C24.52D6

Smallest permutation representation of C24.52D6
On 48 points
Generators in S48
```(2 8)(4 10)(6 12)(13 19)(15 21)(17 23)(25 44)(26 39)(27 46)(28 41)(29 48)(30 43)(31 38)(32 45)(33 40)(34 47)(35 42)(36 37)
(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 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)
(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 30 7 36)(2 35 8 29)(3 28 9 34)(4 33 10 27)(5 26 11 32)(6 31 12 25)(13 42 19 48)(14 47 20 41)(15 40 21 46)(16 45 22 39)(17 38 23 44)(18 43 24 37)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 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 S3 D4 D6 D6 D6 C3⋊D4 2+ 1+4 D4⋊6D6 kernel C24.52D6 C23.26D6 C23.12D6 D6⋊3D4 C12⋊3D4 C24⋊4S3 C2×C4○D12 D4×C2×C6 C22×D4 C2×C12 C22×C4 C2×D4 C24 C2×C4 C6 C2 # reps 1 1 2 4 2 4 1 1 1 4 1 4 2 8 2 4

Matrix representation of C24.52D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 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 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 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 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
,
 10 0 0 0 0 0 0 4 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 0 4 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,10,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0] >;`

C24.52D6 in GAP, Magma, Sage, TeX

`C_2^4._{52}D_6`
`% in TeX`

`G:=Group("C2^4.52D6");`
`// GroupNames label`

`G:=SmallGroup(192,1364);`
`// by ID`

`G=gap.SmallGroup(192,1364);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,6278]);`
`// 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,f*a*f^-1=a*c*d,b*c=c*b,f*b*f^-1=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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