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G = D92  order 324 = 22·34

Direct product of D9 and D9

direct product, metabelian, supersoluble, monomial, A-group

Aliases: D92, C9⋊1D18, 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 2A2B2C3A3B3C6A6B9A···9F9G···9U18A···18F
order1222333669···99···918···18
size1998122418182···24···418···18

36 irreducible representations

dim1112222444
type++++++++++
imageC1C2C2S3D6D9D18S32S3×D9D92
kernelD92C9×D9C9⋊D9C3×D9C3×C9D9C9C32C3C1
# reps1212266169

Matrix representation of D92 ►in GL4(𝔽19) generated by

1000
0100
00714
0052
,
18000
01800
0052
00714
,
51200
71700
0010
0001
,
12200
14700
00180
00018
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
 

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