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G = C10×D11order 220 = 22·5·11

Direct product of C10 and D11

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

Aliases: C10×D11, C1102C2, C223C10, C553C22, C113(C2×C10), SmallGroup(220,12)

Series: Derived Chief Lower central Upper central

C1C11 — C10×D11
C1C11C55C5×D11 — C10×D11
C11 — C10×D11
C1C10

Generators and relations for C10×D11
 G = < a,b,c | a10=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C2
11C22
11C10
11C10
11C2×C10

Smallest permutation representation of C10×D11
On 110 points
Generators in S110
(1 87 54 76 43 65 32 109 21 98)(2 88 55 77 44 66 33 110 22 99)(3 78 45 67 34 56 23 100 12 89)(4 79 46 68 35 57 24 101 13 90)(5 80 47 69 36 58 25 102 14 91)(6 81 48 70 37 59 26 103 15 92)(7 82 49 71 38 60 27 104 16 93)(8 83 50 72 39 61 28 105 17 94)(9 84 51 73 40 62 29 106 18 95)(10 85 52 74 41 63 30 107 19 96)(11 86 53 75 42 64 31 108 20 97)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 66)(11 65)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 77)(20 76)(21 75)(22 74)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 88)(31 87)(32 86)(33 85)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 99)(42 98)(43 97)(44 96)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 110)(53 109)(54 108)(55 107)

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

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

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

C10×D11 is a maximal subgroup of   C55⋊D4  C5⋊D44

70 conjugacy classes

class 1 2A2B2C5A5B5C5D10A10B10C10D10E···10L11A···11E22A···22E55A···55T110A···110T
order122255551010101010···1011···1122···2255···55110···110
size1111111111111111···112···22···22···22···2

70 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D11D22C5×D11C10×D11
kernelC10×D11C5×D11C110D22D11C22C10C5C2C1
# reps121484552020

Matrix representation of C10×D11 in GL3(𝔽331) generated by

20700
0640
0064
,
100
001
0330200
,
100
001
010
G:=sub<GL(3,GF(331))| [207,0,0,0,64,0,0,0,64],[1,0,0,0,0,330,0,1,200],[1,0,0,0,0,1,0,1,0] >;

C10×D11 in GAP, Magma, Sage, TeX

C_{10}\times D_{11}
% in TeX

G:=Group("C10xD11");
// GroupNames label

G:=SmallGroup(220,12);
// by ID

G=gap.SmallGroup(220,12);
# by ID

G:=PCGroup([4,-2,-2,-5,-11,3203]);
// Polycyclic

G:=Group<a,b,c|a^10=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×D11 in TeX

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