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## G = D12⋊19D4order 192 = 26·3

### 7th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D12⋊19D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊19D4
 Lower central C3 — C2×C6 — D12⋊19D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for D1219D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=dbd=a6b, dcd=c-1 >

Subgroups: 1408 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×2], C4 [×7], C22, C22 [×2], C22 [×42], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×11], D4 [×34], C23, C23 [×2], C23 [×25], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×3], D6 [×6], D6 [×28], C2×C6, C2×C6 [×2], C2×C6 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×29], C24 [×4], C4×S3 [×6], D12 [×4], D12 [×10], C2×Dic3 [×3], C3⋊D4 [×4], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3, C22×S3 [×4], C22×S3 [×20], C22×C6, C22×C6 [×2], C4×D4 [×2], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C41D4, C22×D4 [×4], C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4 [×4], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×4], C2×D12 [×4], S3×D4 [×12], C2×C3⋊D4, C2×C3⋊D4 [×6], C22×C12, C6×D4, C6×D4 [×2], S3×C23 [×4], D42, D6⋊D4 [×2], Dic3⋊D4 [×2], Dic35D4, C12⋊D4, C4×C3⋊D4, C232D6 [×2], C123D4, C3×C4⋊D4, C22×D12, C2×S3×D4, C2×S3×D4 [×2], D1219D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C24, C22×S3 [×7], C22×D4 [×2], 2+ 1+4, S3×D4 [×4], S3×C23, D42, C2×S3×D4 [×2], D4○D12, D1219D4

Smallest permutation representation of D1219D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
(1 38 35 16)(2 45 36 23)(3 40 25 18)(4 47 26 13)(5 42 27 20)(6 37 28 15)(7 44 29 22)(8 39 30 17)(9 46 31 24)(10 41 32 19)(11 48 33 14)(12 43 34 21)
(1 29)(2 36)(3 31)(4 26)(5 33)(6 28)(7 35)(8 30)(9 25)(10 32)(11 27)(12 34)(14 20)(16 22)(18 24)(38 44)(40 46)(42 48)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H ··· 2M 2N 2O 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 ··· 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 2 2 4 4 6 ··· 6 12 12 2 2 2 4 4 4 6 6 12 12 2 2 2 4 4 8 8 4 4 4 4 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 2+ 1+4 S3×D4 S3×D4 D4○D12 kernel D12⋊19D4 D6⋊D4 Dic3⋊D4 Dic3⋊5D4 C12⋊D4 C4×C3⋊D4 C23⋊2D6 C12⋊3D4 C3×C4⋊D4 C22×D12 C2×S3×D4 C4⋊D4 D12 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C4 C22 C2 # reps 1 2 2 1 1 1 2 1 1 1 3 1 4 4 2 1 1 3 1 2 2 2

Matrix representation of D1219D4 in GL6(ℤ)

 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1

`G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;`

D1219D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{19}D_4`
`% in TeX`

`G:=Group("D12:19D4");`
`// GroupNames label`

`G:=SmallGroup(192,1168);`
`// by ID`

`G=gap.SmallGroup(192,1168);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,192,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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