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G = D5×C22order 220 = 22·5·11

Direct product of C22 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — D5×C22
C1C5C55D5×C11 — D5×C22
C5 — D5×C22
C1C22

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

5C2
5C2
5C22
5C22
5C22
5C2×C22

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 2A2B2C5A5B10A10B11A···11J22A···22J22K···22AD55A···55T110A···110T
order122255101011···1122···2222···2255···55110···110
size115522221···11···15···52···22···2

88 irreducible representations

dim1111112222
type+++++
imageC1C2C2C11C22C22D5D10D5×C11D5×C22
kernelD5×C22D5×C11C110D10D5C10C22C11C2C1
# reps121102010222020

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

33000
01110
00111
,
100
0215216
0330330
,
100
0330115
001
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

Export

Subgroup lattice of D5×C22 in TeX

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