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## G = D12⋊1D4order 192 = 26·3

### 1st semidirect product of D12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊1D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D4⋊6D6 — D12⋊1D4
 Lower central C3 — C6 — C2×C12 — D12⋊1D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D121D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a7b, dcd=c-1 >

Subgroups: 656 in 168 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3 [×3], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×5], D4 [×16], Q8 [×2], C23 [×2], C23 [×3], Dic3 [×4], C12 [×2], D6 [×7], C2×C6, C2×C6 [×4], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], D12 [×2], C2×Dic3 [×3], C3⋊D4 [×10], C2×C12, C3×D4 [×2], C22×S3 [×3], C22×C6 [×2], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C24⋊C2 [×2], D24 [×2], C4×Dic3, C3×M4(2) [×2], C2×D12, C4○D12 [×2], S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×4], C6×D4, D44D4, D12⋊C4 [×2], C3×C4.D4, C8⋊D6 [×2], C123D4, D46D6, D121D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D44D4, D6⋊D4, D121D4

Character table of D121D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 8A 8B 12A 12B 24A 24B 24C 24D size 1 1 2 4 4 12 12 24 2 2 2 12 12 12 12 2 4 8 8 8 8 4 4 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 linear of order 2 ρ8 1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 linear of order 2 ρ9 2 2 -2 0 0 2 0 0 2 -2 2 0 0 -2 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 0 0 0 -1 2 2 0 0 0 0 -1 -1 1 1 2 -2 -1 -1 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 0 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 -2 -2 0 0 0 -1 2 2 0 0 0 0 -1 -1 1 1 -2 2 -1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 2 2 -2 2 0 0 0 2 -2 -2 0 0 0 0 2 2 2 -2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 0 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 -2 -2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ15 2 2 2 2 -2 0 0 0 2 -2 -2 0 0 0 0 2 2 -2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 -2 0 0 -2 0 0 2 -2 2 0 0 2 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 0 0 0 2 0 2 2 -2 0 -2 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 0 0 -2 0 2 2 -2 0 2 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 1 -1 0 0 1 1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ20 2 2 2 -2 2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 0 0 1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ21 2 2 2 -2 2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 0 0 1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ22 2 2 2 2 -2 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 1 -1 0 0 1 1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 4 0 0 -2 0 0 2 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ24 4 4 -4 0 0 0 0 0 -2 4 -4 0 0 0 0 -2 2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 0 0 0 0 0 -2 -4 4 0 0 0 0 -2 2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 4 0 0 2 0 0 -2 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ27 8 -8 0 0 0 0 0 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D121D4
On 24 points - transitive group 24T343
Generators in S24
```(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(2 6)(3 11)(5 9)(8 12)(13 22 19 16)(14 15 20 21)(17 18 23 24)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)```

`G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (2,6)(3,11)(5,9)(8,12)(13,22,19,16)(14,15,20,21)(17,18,23,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)>;`

`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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (2,6)(3,11)(5,9)(8,12)(13,22,19,16)(14,15,20,21)(17,18,23,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24) );`

`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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)], [(2,6),(3,11),(5,9),(8,12),(13,22,19,16),(14,15,20,21),(17,18,23,24)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24)])`

`G:=TransitiveGroup(24,343);`

Matrix representation of D121D4 in GL6(𝔽73)

 1 1 0 0 0 0 72 0 0 0 0 0 0 0 72 25 0 0 0 0 70 1 0 0 0 0 11 17 0 1 0 0 62 0 72 0
,
 66 7 0 0 0 0 14 7 0 0 0 0 0 0 72 42 71 0 0 0 0 52 70 3 0 0 0 49 11 63 0 0 0 24 63 11
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 48 0 0 0 0 3 72 0 0 0 0 62 49 72 0 0 0 10 49 0 72
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 70 1 0 0 0 0 11 66 0 1 0 0 63 7 1 0

`G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,70,11,62,0,0,25,1,17,0,0,0,0,0,0,72,0,0,0,0,1,0],[66,14,0,0,0,0,7,7,0,0,0,0,0,0,72,0,0,0,0,0,42,52,49,24,0,0,71,70,11,63,0,0,0,3,63,11],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,3,62,10,0,0,48,72,49,49,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,70,11,63,0,0,0,1,66,7,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D121D4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_1D_4`
`% in TeX`

`G:=Group("D12:1D4");`
`// GroupNames label`

`G:=SmallGroup(192,306);`
`// by ID`

`G=gap.SmallGroup(192,306);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,1123,570,136,1684,438,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^7*b,d*c*d=c^-1>;`
`// generators/relations`

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