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## G = C32×D11order 198 = 2·32·11

### Direct product of C32 and D11

Aliases: C32×D11, C332C6, C11⋊(C3×C6), (C3×C33)⋊3C2, SmallGroup(198,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C32×D11
 Chief series C1 — C11 — C33 — C3×C33 — C32×D11
 Lower central C11 — C32×D11
 Upper central C1 — C32

Generators and relations for C32×D11
G = < a,b,c,d | a3=b3=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C32×D11
On 99 points
Generators in S99
(1 98 54)(2 99 55)(3 89 45)(4 90 46)(5 91 47)(6 92 48)(7 93 49)(8 94 50)(9 95 51)(10 96 52)(11 97 53)(12 67 56)(13 68 57)(14 69 58)(15 70 59)(16 71 60)(17 72 61)(18 73 62)(19 74 63)(20 75 64)(21 76 65)(22 77 66)(23 78 34)(24 79 35)(25 80 36)(26 81 37)(27 82 38)(28 83 39)(29 84 40)(30 85 41)(31 86 42)(32 87 43)(33 88 44)
(1 32 21)(2 33 22)(3 23 12)(4 24 13)(5 25 14)(6 26 15)(7 27 16)(8 28 17)(9 29 18)(10 30 19)(11 31 20)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(67 89 78)(68 90 79)(69 91 80)(70 92 81)(71 93 82)(72 94 83)(73 95 84)(74 96 85)(75 97 86)(76 98 87)(77 99 88)
(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)| (1,98,54)(2,99,55)(3,89,45)(4,90,46)(5,91,47)(6,92,48)(7,93,49)(8,94,50)(9,95,51)(10,96,52)(11,97,53)(12,67,56)(13,68,57)(14,69,58)(15,70,59)(16,71,60)(17,72,61)(18,73,62)(19,74,63)(20,75,64)(21,76,65)(22,77,66)(23,78,34)(24,79,35)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44), (1,32,21)(2,33,22)(3,23,12)(4,24,13)(5,25,14)(6,26,15)(7,27,16)(8,28,17)(9,29,18)(10,30,19)(11,31,20)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (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( (1,98,54)(2,99,55)(3,89,45)(4,90,46)(5,91,47)(6,92,48)(7,93,49)(8,94,50)(9,95,51)(10,96,52)(11,97,53)(12,67,56)(13,68,57)(14,69,58)(15,70,59)(16,71,60)(17,72,61)(18,73,62)(19,74,63)(20,75,64)(21,76,65)(22,77,66)(23,78,34)(24,79,35)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44), (1,32,21)(2,33,22)(3,23,12)(4,24,13)(5,25,14)(6,26,15)(7,27,16)(8,28,17)(9,29,18)(10,30,19)(11,31,20)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (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([(1,98,54),(2,99,55),(3,89,45),(4,90,46),(5,91,47),(6,92,48),(7,93,49),(8,94,50),(9,95,51),(10,96,52),(11,97,53),(12,67,56),(13,68,57),(14,69,58),(15,70,59),(16,71,60),(17,72,61),(18,73,62),(19,74,63),(20,75,64),(21,76,65),(22,77,66),(23,78,34),(24,79,35),(25,80,36),(26,81,37),(27,82,38),(28,83,39),(29,84,40),(30,85,41),(31,86,42),(32,87,43),(33,88,44)], [(1,32,21),(2,33,22),(3,23,12),(4,24,13),(5,25,14),(6,26,15),(7,27,16),(8,28,17),(9,29,18),(10,30,19),(11,31,20),(34,56,45),(35,57,46),(36,58,47),(37,59,48),(38,60,49),(39,61,50),(40,62,51),(41,63,52),(42,64,53),(43,65,54),(44,66,55),(67,89,78),(68,90,79),(69,91,80),(70,92,81),(71,93,82),(72,94,83),(73,95,84),(74,96,85),(75,97,86),(76,98,87),(77,99,88)], [(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 ··· 3H 6A ··· 6H 11A ··· 11E 33A ··· 33AN order 1 2 3 ··· 3 6 ··· 6 11 ··· 11 33 ··· 33 size 1 11 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D11 C3×D11 kernel C32×D11 C3×C33 C3×D11 C33 C32 C3 # reps 1 1 8 8 5 40

Matrix representation of C32×D11 in GL3(𝔽67) generated by

 29 0 0 0 1 0 0 0 1
,
 29 0 0 0 37 0 0 0 37
,
 1 0 0 0 13 1 0 56 25
,
 66 0 0 0 58 31 0 32 9
G:=sub<GL(3,GF(67))| [29,0,0,0,1,0,0,0,1],[29,0,0,0,37,0,0,0,37],[1,0,0,0,13,56,0,1,25],[66,0,0,0,58,32,0,31,9] >;

C32×D11 in GAP, Magma, Sage, TeX

C_3^2\times D_{11}
% in TeX

G:=Group("C3^2xD11");
// GroupNames label

G:=SmallGroup(198,5);
// by ID

G=gap.SmallGroup(198,5);
# by ID

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

G:=Group<a,b,c,d|a^3=b^3=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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