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## G = C24.38D6order 192 = 26·3

### 27th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.38D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C24.38D6
 Lower central C3 — C2×C6 — C24.38D6
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C24.38D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 952 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, D45D4, Dic3.D4, S3×C22⋊C4, Dic34D4, D6⋊D4, C23.9D6, Dic3⋊D4, C23.11D6, C4×C3⋊D4, C23.28D6, C127D4, C244S3, C6×C22⋊C4, C2×C4○D12, C22×C3⋊D4, C24.38D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C4○D12, S3×D4, S3×C23, D45D4, C2×C4○D12, C2×S3×D4, D46D6, C24.38D6

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 2 2 2 2 4 6 6 12 12 2 2 2 2 2 4 4 6 6 12 12 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 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 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C4○D12 2+ 1+4 S3×D4 D4⋊6D6 kernel C24.38D6 Dic3.D4 S3×C22⋊C4 Dic3⋊4D4 D6⋊D4 C23.9D6 Dic3⋊D4 C23.11D6 C4×C3⋊D4 C23.28D6 C12⋊7D4 C24⋊4S3 C6×C22⋊C4 C2×C4○D12 C22×C3⋊D4 C2×C22⋊C4 C3⋊D4 C22⋊C4 C22×C4 C24 C2×C6 C22 C6 C22 C2 # reps 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 4 4 2 1 4 8 1 2 2

Matrix representation of C24.38D6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 5 12
,
 2 9 0 0 4 11 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
,
 0 5 0 0 8 5 0 0 0 0 12 3 0 0 0 1
,
 5 0 0 0 5 8 0 0 0 0 12 3 0 0 0 1
`G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,5,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[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],[0,8,0,0,5,5,0,0,0,0,12,0,0,0,3,1],[5,5,0,0,0,8,0,0,0,0,12,0,0,0,3,1] >;`

C24.38D6 in GAP, Magma, Sage, TeX

`C_2^4._{38}D_6`
`% in TeX`

`G:=Group("C2^4.38D6");`
`// GroupNames label`

`G:=SmallGroup(192,1049);`
`// by ID`

`G=gap.SmallGroup(192,1049);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,6278]);`
`// Polycyclic`

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