Copied to
clipboard

## G = D92order 324 = 22·34

### Direct product of D9 and D9

Aliases: D92, C91D18, C92⋊C22, C9⋊D9⋊C2, (C9×D9)⋊C2, (C3×D9).S3, C32.6S32, C3.1(S3×D9), (C3×C9).4D6, SmallGroup(324,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — D92
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×D9 — D92
 Lower central C92 — D92
 Upper central C1

Generators and relations for D92
G = < a,b,c,d | a9=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 589 in 59 conjugacy classes, 17 normal (5 characteristic)
C1, C2, C3, C3, C22, S3, C6, C9, C9, C32, D6, D9, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, D18, S32, C3×D9, S3×C9, C9⋊S3, C92, S3×D9, C9×D9, C9⋊D9, D92
Quotients: C1, C2, C22, S3, D6, D9, D18, S32, S3×D9, D92

Permutation representations of D92
On 18 points - transitive group 18T140
Generators in S18
```(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)
(1 9 8 7 6 5 4 3 2)(10 11 12 13 14 15 16 17 18)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)```

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

`G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14), (1,9,8,7,6,5,4,3,2)(10,11,12,13,14,15,16,17,18), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14)], [(1,9,8,7,6,5,4,3,2),(10,11,12,13,14,15,16,17,18)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,10),(8,11),(9,12)]])`

`G:=TransitiveGroup(18,140);`

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 9A ··· 9F 9G ··· 9U 18A ··· 18F order 1 2 2 2 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 9 81 2 2 4 18 18 2 ··· 2 4 ··· 4 18 ··· 18

36 irreducible representations

 dim 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 S3 D6 D9 D18 S32 S3×D9 D92 kernel D92 C9×D9 C9⋊D9 C3×D9 C3×C9 D9 C9 C32 C3 C1 # reps 1 2 1 2 2 6 6 1 6 9

Matrix representation of D92 in GL4(𝔽19) generated by

 1 0 0 0 0 1 0 0 0 0 7 14 0 0 5 2
,
 18 0 0 0 0 18 0 0 0 0 5 2 0 0 7 14
,
 5 12 0 0 7 17 0 0 0 0 1 0 0 0 0 1
,
 12 2 0 0 14 7 0 0 0 0 18 0 0 0 0 18
`G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,7,5,0,0,14,2],[18,0,0,0,0,18,0,0,0,0,5,7,0,0,2,14],[5,7,0,0,12,17,0,0,0,0,1,0,0,0,0,1],[12,14,0,0,2,7,0,0,0,0,18,0,0,0,0,18] >;`

D92 in GAP, Magma, Sage, TeX

`D_9^2`
`% in TeX`

`G:=Group("D9^2");`
`// GroupNames label`

`G:=SmallGroup(324,36);`
`// by ID`

`G=gap.SmallGroup(324,36);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-3,-3,404,338,3171,453,1090,7781]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽