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G = C10xD11order 220 = 22·5·11

Direct product of C10 and D11

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

Aliases: C10xD11, C110:2C2, C22:3C10, C55:3C22, C11:3(C2xC10), SmallGroup(220,12)

Series: Derived Chief Lower central Upper central

C1C11 — C10xD11
C1C11C55C5xD11 — C10xD11
C11 — C10xD11
C1C10

Generators and relations for C10xD11
 G = < a,b,c | a10=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 80 in 20 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C22, C5, C10, C2xC10, D11, D22, C5xD11, C10xD11
11C2
11C2
11C22
11C10
11C10
11C2xC10

Smallest permutation representation of C10xD11
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)]])

C10xD11 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+++++
imageC1C2C2C5C10C10D11D22C5xD11C10xD11
kernelC10xD11C5xD11C110D22D11C22C10C5C2C1
# reps121484552020

Matrix representation of C10xD11 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] >;

C10xD11 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 C10xD11 in TeX

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