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G = C9×D11order 198 = 2·32·11

Direct product of C9 and D11

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

Aliases: C9×D11, C11⋊C18, C992C2, C33.C6, C3.(C3×D11), (C3×D11).C3, SmallGroup(198,2)

Series: Derived Chief Lower central Upper central

C1C11 — C9×D11
C1C11C33C99 — C9×D11
C11 — C9×D11
C1C9

Generators and relations for C9×D11
 G = < a,b,c | a9=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C6
11C18

Smallest permutation representation of C9×D11
On 99 points
Generators in S99
(1 98 65 32 87 54 21 76 43)(2 99 66 33 88 55 22 77 44)(3 89 56 23 78 45 12 67 34)(4 90 57 24 79 46 13 68 35)(5 91 58 25 80 47 14 69 36)(6 92 59 26 81 48 15 70 37)(7 93 60 27 82 49 16 71 38)(8 94 61 28 83 50 17 72 39)(9 95 62 29 84 51 18 73 40)(10 96 63 30 85 52 19 74 41)(11 97 64 31 86 53 20 75 42)
(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)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)

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

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

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

63 conjugacy classes

class 1  2 3A3B6A6B9A···9F11A···11E18A···18F33A···33J99A···99AD
order1233669···911···1118···1833···3399···99
size1111111111···12···211···112···22···2

63 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D11C3×D11C9×D11
kernelC9×D11C99C3×D11C33D11C11C9C3C1
# reps11226651030

Matrix representation of C9×D11 in GL2(𝔽199) generated by

1780
0178
,
1231
1980
,
01
10
G:=sub<GL(2,GF(199))| [178,0,0,178],[123,198,1,0],[0,1,1,0] >;

C9×D11 in GAP, Magma, Sage, TeX

C_9\times D_{11}
% in TeX

G:=Group("C9xD11");
// GroupNames label

G:=SmallGroup(198,2);
// by ID

G=gap.SmallGroup(198,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-11,29,2883]);
// Polycyclic

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

Export

Subgroup lattice of C9×D11 in TeX

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