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## G = D4.11D12order 192 = 26·3

### 1st non-split extension by D4 of D12 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4.11D12
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — D4○D12 — D4.11D12
 Lower central C3 — C6 — C12 — D4.11D12
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.11D12
G = < a,b,c,d | a4=b2=1, c12=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >

Subgroups: 752 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C24, Dic6, Dic6, Dic6, C4×S3, D12, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C24⋊C2, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×Dic6, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, D4○SD16, C2×C24⋊C2, C4○D24, C8⋊D6, C8.D6, C3×C8○D4, D4○D12, Q8○D12, D4.11D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, D4○SD16, C22×D12, D4.11D12

Smallest permutation representation of D4.11D12
On 48 points
Generators in S48
```(1 25 13 37)(2 26 14 38)(3 27 15 39)(4 28 16 40)(5 29 17 41)(6 30 18 42)(7 31 19 43)(8 32 20 44)(9 33 21 45)(10 34 22 46)(11 35 23 47)(12 36 24 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
(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 13 24)(2 23 14 11)(3 10 15 22)(4 21 16 9)(5 8 17 20)(6 19 18 7)(25 36 37 48)(26 47 38 35)(27 34 39 46)(28 45 40 33)(29 32 41 44)(30 43 42 31)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 24A 24B 24C 24D 24E ··· 24J order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 8 8 8 8 8 12 12 12 12 12 24 24 24 24 24 ··· 24 size 1 1 2 2 2 12 12 12 12 2 2 2 2 2 12 12 12 12 2 4 4 4 2 2 4 4 4 2 2 4 4 4 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 D12 D4○SD16 D4.11D12 kernel D4.11D12 C2×C24⋊C2 C4○D24 C8⋊D6 C8.D6 C3×C8○D4 D4○D12 Q8○D12 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 3 3 1 1 1 1 3 1 3 3 1 6 2 2 4

Matrix representation of D4.11D12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 63 11 22 72 0 0 30 44 48 72 0 0 70 3 40 0 0 0 0 2 20 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 10 62 3 0 0 0 43 29 25 0 0 0 3 70 33 0 0 0 0 71 53 1
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 67 6 9 7 0 0 67 67 35 34 0 0 0 0 0 55 0 0 0 0 4 61
,
 1 1 0 0 0 0 0 72 0 0 0 0 0 0 67 6 38 66 0 0 6 6 12 39 0 0 0 0 61 18 0 0 0 0 69 12

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,63,30,70,0,0,0,11,44,3,2,0,0,22,48,40,20,0,0,72,72,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,10,43,3,0,0,0,62,29,70,71,0,0,3,25,33,53,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,9,35,0,4,0,0,7,34,55,61],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,38,12,61,69,0,0,66,39,18,12] >;`

D4.11D12 in GAP, Magma, Sage, TeX

`D_4._{11}D_{12}`
`% in TeX`

`G:=Group("D4.11D12");`
`// GroupNames label`

`G:=SmallGroup(192,1310);`
`// by ID`

`G=gap.SmallGroup(192,1310);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,80,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^12=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;`
`// generators/relations`

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