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G = D9xD11order 396 = 22·32·11

Direct product of D9 and D11

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

Aliases: D9xD11, D99:C2, C9:1D22, C99:C22, C33.D6, C11:1D18, (C9xD11):C2, (C11xD9):C2, C3.(S3xD11), (C3xD11).S3, SmallGroup(396,5)

Series: Derived Chief Lower central Upper central

C1C99 — D9xD11
C1C3C33C99C9xD11 — D9xD11
C99 — D9xD11
C1

Generators and relations for D9xD11
 G = < a,b,c,d | a9=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 380 in 30 conjugacy classes, 13 normal (all characteristic)
Quotients: C1, C2, C22, S3, D6, D9, D11, D18, D22, S3xD11, D9xD11
9C2
11C2
99C2
99C22
3S3
11C6
33S3
9C22
9D11
33D6
11C18
11D9
9D22
3S3xC11
3D33
11D18
3S3xD11

Smallest permutation representation of D9xD11
On 99 points
Generators in S99
(1 91 58 35 30 86 53 18 68)(2 92 59 36 31 87 54 19 69)(3 93 60 37 32 88 55 20 70)(4 94 61 38 33 78 45 21 71)(5 95 62 39 23 79 46 22 72)(6 96 63 40 24 80 47 12 73)(7 97 64 41 25 81 48 13 74)(8 98 65 42 26 82 49 14 75)(9 99 66 43 27 83 50 15 76)(10 89 56 44 28 84 51 16 77)(11 90 57 34 29 85 52 17 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 67)(12 96)(13 97)(14 98)(15 99)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(34 85)(35 86)(36 87)(37 88)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 56)(52 57)(53 58)(54 59)(55 60)
(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 11)(2 10)(3 9)(4 8)(5 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 25)(26 33)(27 32)(28 31)(29 30)(34 35)(36 44)(37 43)(38 42)(39 41)(45 49)(46 48)(50 55)(51 54)(52 53)(56 59)(57 58)(60 66)(61 65)(62 64)(67 68)(69 77)(70 76)(71 75)(72 74)(78 82)(79 81)(83 88)(84 87)(85 86)(89 92)(90 91)(93 99)(94 98)(95 97)

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

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

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

42 conjugacy classes

class 1 2A2B2C 3  6 9A9B9C11A···11E18A18B18C22A···22E33A···33E99A···99O
order12223699911···1118181822···2233···3399···99
size1911992222222···222222218···184···44···4

42 irreducible representations

dim111122222244
type++++++++++++
imageC1C2C2C2S3D6D9D11D18D22S3xD11D9xD11
kernelD9xD11C11xD9C9xD11D99C3xD11C33D11D9C11C9C3C1
# reps1111113535515

Matrix representation of D9xD11 in GL4(F199) generated by

5710800
9114800
0010
0001
,
5710800
5114200
0010
0001
,
1000
0100
0001
001983
,
1000
0100
0001
0010
G:=sub<GL(4,GF(199))| [57,91,0,0,108,148,0,0,0,0,1,0,0,0,0,1],[57,51,0,0,108,142,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,198,0,0,1,3],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

D9xD11 in GAP, Magma, Sage, TeX

D_9\times D_{11}
% in TeX

G:=Group("D9xD11");
// GroupNames label

G:=SmallGroup(396,5);
// by ID

G=gap.SmallGroup(396,5);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-3,1057,1002,2403,3309]);
// Polycyclic

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

Export

Subgroup lattice of D9xD11 in TeX

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