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## G = C9×D11order 198 = 2·32·11

### Direct product of C9 and D11

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C9×D11
 Chief series C1 — C11 — C33 — C99 — C9×D11
 Lower central C11 — C9×D11
 Upper central C1 — C9

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

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 3A 3B 6A 6B 9A ··· 9F 11A ··· 11E 18A ··· 18F 33A ··· 33J 99A ··· 99AD order 1 2 3 3 6 6 9 ··· 9 11 ··· 11 18 ··· 18 33 ··· 33 99 ··· 99 size 1 11 1 1 11 11 1 ··· 1 2 ··· 2 11 ··· 11 2 ··· 2 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 D11 C3×D11 C9×D11 kernel C9×D11 C99 C3×D11 C33 D11 C11 C9 C3 C1 # reps 1 1 2 2 6 6 5 10 30

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

 178 0 0 178
,
 123 1 198 0
,
 0 1 1 0
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

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