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## G = D12⋊18D6order 288 = 25·32

### 2nd semidirect product of D12 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊18D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C3⋊D24 — D12⋊18D6
 Lower central C32 — C3×C6 — C3×C12 — D12⋊18D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D1218D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=ab, dcd=c-1 >

Subgroups: 866 in 163 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×9], C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×17], C2×C6 [×2], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, D12 [×10], C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×4], C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6, C2×C3⋊S3 [×4], C62, C24⋊C2 [×2], D24 [×2], C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C3×M4(2), C2×D12 [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3 [×2], C12⋊S3, C6×C12, C22×C3⋊S3, C8⋊D6, D4⋊D6, C3⋊D24 [×2], C325SD16 [×2], C3×C4.Dic3, C3×C4○D12, C2×C12⋊S3, D1218D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C8⋊D6, D4⋊D6, C2×C3⋊D12, D1218D6

Permutation representations of D1218D6
On 24 points - transitive group 24T613
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)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)```

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

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

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

`G:=TransitiveGroup(24,613);`

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C ··· 6G 6H 6I 8A 8B 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 12 36 36 2 2 4 2 2 12 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12 12 12

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D12 C3⋊D4 D12 C3⋊D4 C8⋊C22 S32 C3⋊D12 C2×S32 C3⋊D12 C8⋊D6 D4⋊D6 D12⋊18D6 kernel D12⋊18D6 C3⋊D24 C32⋊5SD16 C3×C4.Dic3 C3×C4○D12 C2×C12⋊S3 C4.Dic3 C4○D12 C3×C12 C62 C3⋊C8 Dic6 D12 C2×C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 1 1 1 1 1 2 2 4

Matrix representation of D1218D6 in GL4(𝔽73) generated by

 14 66 0 0 7 7 0 0 0 0 14 66 0 0 7 7
,
 0 0 66 66 0 0 59 7 66 66 0 0 59 7 0 0
,
 0 1 0 0 72 1 0 0 0 0 72 1 0 0 72 0
,
 1 72 0 0 0 72 0 0 0 0 66 14 0 0 7 7
`G:=sub<GL(4,GF(73))| [14,7,0,0,66,7,0,0,0,0,14,7,0,0,66,7],[0,0,66,59,0,0,66,7,66,59,0,0,66,7,0,0],[0,72,0,0,1,1,0,0,0,0,72,72,0,0,1,0],[1,0,0,0,72,72,0,0,0,0,66,7,0,0,14,7] >;`

D1218D6 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_{18}D_6`
`% in TeX`

`G:=Group("D12:18D6");`
`// GroupNames label`

`G:=SmallGroup(288,473);`
`// by ID`

`G=gap.SmallGroup(288,473);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,219,100,675,80,1356,9414]);`
`// Polycyclic`

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

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