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## G = D5×C22order 220 = 22·5·11

### Direct product of C22 and D5

Aliases: D5×C22, C10⋊C22, C1103C2, C554C22, C5⋊(C2×C22), SmallGroup(220,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C22
 Chief series C1 — C5 — C55 — D5×C11 — D5×C22
 Lower central C5 — D5×C22
 Upper central C1 — C22

Generators and relations for D5×C22
G = < a,b,c | a22=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C22
On 110 points
Generators in S110
(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 56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110)
(1 88 94 58 39)(2 67 95 59 40)(3 68 96 60 41)(4 69 97 61 42)(5 70 98 62 43)(6 71 99 63 44)(7 72 100 64 23)(8 73 101 65 24)(9 74 102 66 25)(10 75 103 45 26)(11 76 104 46 27)(12 77 105 47 28)(13 78 106 48 29)(14 79 107 49 30)(15 80 108 50 31)(16 81 109 51 32)(17 82 110 52 33)(18 83 89 53 34)(19 84 90 54 35)(20 85 91 55 36)(21 86 92 56 37)(22 87 93 57 38)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 23)(19 24)(20 25)(21 26)(22 27)(45 86)(46 87)(47 88)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)

G:=sub<Sym(110)| (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,88,94,58,39)(2,67,95,59,40)(3,68,96,60,41)(4,69,97,61,42)(5,70,98,62,43)(6,71,99,63,44)(7,72,100,64,23)(8,73,101,65,24)(9,74,102,66,25)(10,75,103,45,26)(11,76,104,46,27)(12,77,105,47,28)(13,78,106,48,29)(14,79,107,49,30)(15,80,108,50,31)(16,81,109,51,32)(17,82,110,52,33)(18,83,89,53,34)(19,84,90,54,35)(20,85,91,55,36)(21,86,92,56,37)(22,87,93,57,38), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,23)(19,24)(20,25)(21,26)(22,27)(45,86)(46,87)(47,88)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)>;

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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,88,94,58,39)(2,67,95,59,40)(3,68,96,60,41)(4,69,97,61,42)(5,70,98,62,43)(6,71,99,63,44)(7,72,100,64,23)(8,73,101,65,24)(9,74,102,66,25)(10,75,103,45,26)(11,76,104,46,27)(12,77,105,47,28)(13,78,106,48,29)(14,79,107,49,30)(15,80,108,50,31)(16,81,109,51,32)(17,82,110,52,33)(18,83,89,53,34)(19,84,90,54,35)(20,85,91,55,36)(21,86,92,56,37)(22,87,93,57,38), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,23)(19,24)(20,25)(21,26)(22,27)(45,86)(46,87)(47,88)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110) );

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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)], [(1,88,94,58,39),(2,67,95,59,40),(3,68,96,60,41),(4,69,97,61,42),(5,70,98,62,43),(6,71,99,63,44),(7,72,100,64,23),(8,73,101,65,24),(9,74,102,66,25),(10,75,103,45,26),(11,76,104,46,27),(12,77,105,47,28),(13,78,106,48,29),(14,79,107,49,30),(15,80,108,50,31),(16,81,109,51,32),(17,82,110,52,33),(18,83,89,53,34),(19,84,90,54,35),(20,85,91,55,36),(21,86,92,56,37),(22,87,93,57,38)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,23),(19,24),(20,25),(21,26),(22,27),(45,86),(46,87),(47,88),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110)])

D5×C22 is a maximal subgroup of   C55⋊D4  C11⋊D20

88 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 11A ··· 11J 22A ··· 22J 22K ··· 22AD 55A ··· 55T 110A ··· 110T order 1 2 2 2 5 5 10 10 11 ··· 11 22 ··· 22 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 5 5 2 2 2 2 1 ··· 1 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C11 C22 C22 D5 D10 D5×C11 D5×C22 kernel D5×C22 D5×C11 C110 D10 D5 C10 C22 C11 C2 C1 # reps 1 2 1 10 20 10 2 2 20 20

Matrix representation of D5×C22 in GL3(𝔽331) generated by

 330 0 0 0 111 0 0 0 111
,
 1 0 0 0 215 216 0 330 330
,
 1 0 0 0 330 115 0 0 1
G:=sub<GL(3,GF(331))| [330,0,0,0,111,0,0,0,111],[1,0,0,0,215,330,0,216,330],[1,0,0,0,330,0,0,115,1] >;

D5×C22 in GAP, Magma, Sage, TeX

D_5\times C_{22}
% in TeX

G:=Group("D5xC22");
// GroupNames label

G:=SmallGroup(220,13);
// by ID

G=gap.SmallGroup(220,13);
# by ID

G:=PCGroup([4,-2,-2,-11,-5,2819]);
// Polycyclic

G:=Group<a,b,c|a^22=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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