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## G = C11×Dic11order 484 = 22·112

### Direct product of C11 and Dic11

Aliases: C11×Dic11, C11⋊C44, C22.C22, C1122C4, C22.4D11, C2.(C11×D11), (C11×C22).1C2, SmallGroup(484,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C11×Dic11
 Chief series C1 — C11 — C22 — C11×C22 — C11×Dic11
 Lower central C11 — C11×Dic11
 Upper central C1 — C22

Generators and relations for C11×Dic11
G = < a,b,c | a11=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×Dic11
On 44 points
Generators in S44
(1 7 13 19 3 9 15 21 5 11 17)(2 8 14 20 4 10 16 22 6 12 18)(23 39 33 27 43 37 31 25 41 35 29)(24 40 34 28 44 38 32 26 42 36 30)
(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 23 12 34)(2 44 13 33)(3 43 14 32)(4 42 15 31)(5 41 16 30)(6 40 17 29)(7 39 18 28)(8 38 19 27)(9 37 20 26)(10 36 21 25)(11 35 22 24)

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

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

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

154 conjugacy classes

 class 1 2 4A 4B 11A ··· 11J 11K ··· 11BM 22A ··· 22J 22K ··· 22BM 44A ··· 44T order 1 2 4 4 11 ··· 11 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 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 C4 C11 C22 C44 D11 Dic11 C11×D11 C11×Dic11 kernel C11×Dic11 C11×C22 C112 Dic11 C22 C11 C22 C11 C2 C1 # reps 1 1 2 10 10 20 5 5 50 50

Matrix representation of C11×Dic11 in GL2(𝔽23) generated by

 18 0 0 18
,
 21 0 0 11
,
 0 22 1 0
G:=sub<GL(2,GF(23))| [18,0,0,18],[21,0,0,11],[0,1,22,0] >;

C11×Dic11 in GAP, Magma, Sage, TeX

C_{11}\times {\rm Dic}_{11}
% in TeX

G:=Group("C11xDic11");
// GroupNames label

G:=SmallGroup(484,5);
// by ID

G=gap.SmallGroup(484,5);
# by ID

G:=PCGroup([4,-2,-11,-2,-11,88,7043]);
// Polycyclic

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

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