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## G = D5×C11order 110 = 2·5·11

### Direct product of C11 and D5

Aliases: D5×C11, C5⋊C22, C553C2, SmallGroup(110,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C11
 Chief series C1 — C5 — C55 — D5×C11
 Lower central C5 — D5×C11
 Upper central C1 — C11

Generators and relations for D5×C11
G = < a,b,c | a11=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C11
On 55 points
Generators in S55
(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)(45 46 47 48 49 50 51 52 53 54 55)
(1 50 23 16 34)(2 51 24 17 35)(3 52 25 18 36)(4 53 26 19 37)(5 54 27 20 38)(6 55 28 21 39)(7 45 29 22 40)(8 46 30 12 41)(9 47 31 13 42)(10 48 32 14 43)(11 49 33 15 44)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 45)

G:=sub<Sym(55)| (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)(45,46,47,48,49,50,51,52,53,54,55), (1,50,23,16,34)(2,51,24,17,35)(3,52,25,18,36)(4,53,26,19,37)(5,54,27,20,38)(6,55,28,21,39)(7,45,29,22,40)(8,46,30,12,41)(9,47,31,13,42)(10,48,32,14,43)(11,49,33,15,44), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,45)>;

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)(45,46,47,48,49,50,51,52,53,54,55), (1,50,23,16,34)(2,51,24,17,35)(3,52,25,18,36)(4,53,26,19,37)(5,54,27,20,38)(6,55,28,21,39)(7,45,29,22,40)(8,46,30,12,41)(9,47,31,13,42)(10,48,32,14,43)(11,49,33,15,44), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,45) );

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),(45,46,47,48,49,50,51,52,53,54,55)], [(1,50,23,16,34),(2,51,24,17,35),(3,52,25,18,36),(4,53,26,19,37),(5,54,27,20,38),(6,55,28,21,39),(7,45,29,22,40),(8,46,30,12,41),(9,47,31,13,42),(10,48,32,14,43),(11,49,33,15,44)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,45)]])

D5×C11 is a maximal subgroup of   C11⋊F5

44 conjugacy classes

 class 1 2 5A 5B 11A ··· 11J 22A ··· 22J 55A ··· 55T order 1 2 5 5 11 ··· 11 22 ··· 22 55 ··· 55 size 1 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C11 C22 D5 D5×C11 kernel D5×C11 C55 D5 C5 C11 C1 # reps 1 1 10 10 2 20

Matrix representation of D5×C11 in GL2(𝔽331) generated by

 180 0 0 180
,
 330 118 330 117
,
 330 0 330 1
G:=sub<GL(2,GF(331))| [180,0,0,180],[330,330,118,117],[330,330,0,1] >;

D5×C11 in GAP, Magma, Sage, TeX

D_5\times C_{11}
% in TeX

G:=Group("D5xC11");
// GroupNames label

G:=SmallGroup(110,3);
// by ID

G=gap.SmallGroup(110,3);
# by ID

G:=PCGroup([3,-2,-11,-5,794]);
// Polycyclic

G:=Group<a,b,c|a^11=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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