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## G = D11×C22order 484 = 22·112

### Direct product of C22 and D11

Aliases: D11×C22, C22⋊C22, C1122C22, C11⋊(C2×C22), (C11×C22)⋊1C2, SmallGroup(484,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — D11×C22
 Chief series C1 — C11 — C112 — C11×D11 — D11×C22
 Lower central C11 — D11×C22
 Upper central C1 — C22

Generators and relations for D11×C22
G = < a,b,c | a22=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D11×C22
On 44 points
Generators in S44
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22)(23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44)
(1 5 9 13 17 21 3 7 11 15 19)(2 6 10 14 18 22 4 8 12 16 20)(23 41 37 33 29 25 43 39 35 31 27)(24 42 38 34 30 26 44 40 36 32 28)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)

G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,5,9,13,17,21,3,7,11,15,19)(2,6,10,14,18,22,4,8,12,16,20)(23,41,37,33,29,25,43,39,35,31,27)(24,42,38,34,30,26,44,40,36,32,28), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,5,9,13,17,21,3,7,11,15,19)(2,6,10,14,18,22,4,8,12,16,20)(23,41,37,33,29,25,43,39,35,31,27)(24,42,38,34,30,26,44,40,36,32,28), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)], [(1,5,9,13,17,21,3,7,11,15,19),(2,6,10,14,18,22,4,8,12,16,20),(23,41,37,33,29,25,43,39,35,31,27),(24,42,38,34,30,26,44,40,36,32,28)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31)])

154 conjugacy classes

 class 1 2A 2B 2C 11A ··· 11J 11K ··· 11BM 22A ··· 22J 22K ··· 22BM 22BN ··· 22CG order 1 2 2 2 11 ··· 11 11 ··· 11 22 ··· 22 22 ··· 22 22 ··· 22 size 1 1 11 11 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 11 ··· 11

154 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C11 C22 C22 D11 D22 C11×D11 D11×C22 kernel D11×C22 C11×D11 C11×C22 D22 D11 C22 C22 C11 C2 C1 # reps 1 2 1 10 20 10 5 5 50 50

Matrix representation of D11×C22 in GL2(𝔽23) generated by

 19 0 0 19
,
 3 0 0 8
,
 0 8 3 0
G:=sub<GL(2,GF(23))| [19,0,0,19],[3,0,0,8],[0,3,8,0] >;

D11×C22 in GAP, Magma, Sage, TeX

D_{11}\times C_{22}
% in TeX

G:=Group("D11xC22");
// GroupNames label

G:=SmallGroup(484,10);
// by ID

G=gap.SmallGroup(484,10);
# by ID

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

G:=Group<a,b,c|a^22=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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