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G = D112order 484 = 22·112

Direct product of D11 and D11

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

Aliases: D112, C111D22, C112⋊C22, C11⋊D11⋊C2, (C11×D11)⋊C2, SmallGroup(484,9)

Series: Derived Chief Lower central Upper central

C1C112 — D112
C1C11C112C11×D11 — D112
C112 — D112
C1

Generators and relations for D112
 G = < a,b,c,d | a11=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

11C2
11C2
121C2
2C11
2C11
2C11
2C11
2C11
121C22
11D11
11C22
11C22
11D11
22D11
22D11
22D11
22D11
22D11
11D22
11D22

Permutation representations of D112
On 22 points - transitive group 22T9
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(1 12)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)
(1 3 5 7 9 11 2 4 6 8 10)(12 21 19 17 15 13 22 20 18 16 14)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 12)(10 13)(11 14)

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

G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13), (1,3,5,7,9,11,2,4,6,8,10)(12,21,19,17,15,13,22,20,18,16,14), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,12)(10,13)(11,14) );

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

G:=TransitiveGroup(22,9);

49 conjugacy classes

class 1 2A2B2C11A···11J11K···11AI22A···22J
order122211···1111···1122···22
size111111212···24···422···22

49 irreducible representations

dim111224
type++++++
imageC1C2C2D11D22D112
kernelD112C11×D11C11⋊D11D11C11C1
# reps121101025

Matrix representation of D112 in GL4(𝔽23) generated by

1000
0100
001420
00113
,
1000
0100
0010
00822
,
141000
21300
0010
0001
,
16300
7700
00220
00022
G:=sub<GL(4,GF(23))| [1,0,0,0,0,1,0,0,0,0,14,1,0,0,20,13],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,22],[14,2,0,0,10,13,0,0,0,0,1,0,0,0,0,1],[16,7,0,0,3,7,0,0,0,0,22,0,0,0,0,22] >;

D112 in GAP, Magma, Sage, TeX

D_{11}^2
% in TeX

G:=Group("D11^2");
// GroupNames label

G:=SmallGroup(484,9);
// by ID

G=gap.SmallGroup(484,9);
# by ID

G:=PCGroup([4,-2,-2,-11,-11,246,7043]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^2=c^11=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

Export

Subgroup lattice of D112 in TeX

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