Copied to
clipboard

G = C11×D9order 198 = 2·32·11

Direct product of C11 and D9

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

Aliases: C11×D9, C9⋊C22, C993C2, C33.2S3, C3.(S3×C11), SmallGroup(198,1)

Series: Derived Chief Lower central Upper central

C1C9 — C11×D9
C1C3C9C99 — C11×D9
C9 — C11×D9
C1C11

Generators and relations for C11×D9
 G = < a,b,c | a11=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C22
3S3×C11

Smallest permutation representation of C11×D9
On 99 points
Generators in S99
(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)
(1 55 71 21 32 61 44 90 79)(2 45 72 22 33 62 34 91 80)(3 46 73 12 23 63 35 92 81)(4 47 74 13 24 64 36 93 82)(5 48 75 14 25 65 37 94 83)(6 49 76 15 26 66 38 95 84)(7 50 77 16 27 56 39 96 85)(8 51 67 17 28 57 40 97 86)(9 52 68 18 29 58 41 98 87)(10 53 69 19 30 59 42 99 88)(11 54 70 20 31 60 43 89 78)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 78)(12 63)(13 64)(14 65)(15 66)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 67)(41 68)(42 69)(43 70)(44 71)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 89)(55 90)

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

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

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

66 conjugacy classes

class 1  2  3 9A9B9C11A···11J22A···22J33A···33J99A···99AD
order12399911···1122···2233···3399···99
size1922221···19···92···22···2

66 irreducible representations

dim11112222
type++++
imageC1C2C11C22S3D9S3×C11C11×D9
kernelC11×D9C99D9C9C33C11C3C1
# reps111010131030

Matrix representation of C11×D9 in GL2(𝔽199) generated by

1880
0188
,
108142
5751
,
5791
148142
G:=sub<GL(2,GF(199))| [188,0,0,188],[108,57,142,51],[57,148,91,142] >;

C11×D9 in GAP, Magma, Sage, TeX

C_{11}\times D_9
% in TeX

G:=Group("C11xD9");
// GroupNames label

G:=SmallGroup(198,1);
// by ID

G=gap.SmallGroup(198,1);
# by ID

G:=PCGroup([4,-2,-11,-3,-3,1322,82,2115]);
// Polycyclic

G:=Group<a,b,c|a^11=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×D9 in TeX

׿
×
𝔽