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## G = D12.39C23order 192 = 26·3

### 20th non-split extension by D12 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D12.39C23
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4 — D12.39C23
 Lower central C3 — C6 — D12.39C23
 Upper central C1 — C2 — 2- 1+4

Generators and relations for D12.39C23
G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=a-1, ac=ca, ad=da, eae=a5, cbc=a6b, bd=db, ebe=a10b, dcd=ece=a6c, de=ed >

Subgroups: 1704 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, 2- 1+4, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C6×Q8, C3×C4○D4, C2.C25, C2×Q83S3, Q8.15D6, S3×C4○D4, D4○D12, C3×2- 1+4, D12.39C23
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, C25, S3×C23, C2.C25, S3×C24, D12.39C23

Smallest permutation representation of D12.39C23
On 48 points
Generators in S48
```(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 9)(2 8)(3 7)(4 6)(10 12)(13 23)(14 22)(15 21)(16 20)(17 19)(25 35)(26 34)(27 33)(28 32)(29 31)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(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 17)(2 22)(3 15)(4 20)(5 13)(6 18)(7 23)(8 16)(9 21)(10 14)(11 19)(12 24)(25 47)(26 40)(27 45)(28 38)(29 43)(30 48)(31 41)(32 46)(33 39)(34 44)(35 37)(36 42)```

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

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

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

51 conjugacy classes

 class 1 2A 2B ··· 2F 2G ··· 2P 3 4A ··· 4J 4K 4L 4M ··· 4Q 6A 6B ··· 6F 12A ··· 12J order 1 2 2 ··· 2 2 ··· 2 3 4 ··· 4 4 4 4 ··· 4 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 6 ··· 6 2 2 ··· 2 3 3 6 ··· 6 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 C2.C25 D12.39C23 kernel D12.39C23 C2×Q8⋊3S3 Q8.15D6 S3×C4○D4 D4○D12 C3×2- 1+4 2- 1+4 C2×Q8 C4○D4 C3 C1 # reps 1 5 5 10 10 1 1 5 10 2 1

Matrix representation of D12.39C23 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 5 0 0 0 0 10 12 0 0 0 0 0 0 1 5 0 0 0 0 10 12
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 1 5 0 0 0 0 0 12 0 0 0 0 0 0 12 8 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 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
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 8 1 0 0 0 0 2 5 0 0 0 0 0 0 5 12 0 0 0 0 11 8

`G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,10,0,0,0,0,5,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,8,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[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],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5,0,0,0,0,0,0,5,11,0,0,0,0,12,8] >;`

D12.39C23 in GAP, Magma, Sage, TeX

`D_{12}._{39}C_2^3`
`% in TeX`

`G:=Group("D12.39C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,1527);`
`// by ID`

`G=gap.SmallGroup(192,1527);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=a^6*b,b*d=d*b,e*b*e=a^10*b,d*c*d=e*c*e=a^6*c,d*e=e*d>;`
`// generators/relations`

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