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G = C3×C33order 99 = 32·11

Abelian group of type [3,33]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C33, SmallGroup(99,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C33
C1C11C33 — C3×C33
C1 — C3×C33
C1 — C3×C33

Generators and relations for C3×C33
 G = < a,b | a3=b33=1, ab=ba >


Smallest permutation representation of C3×C33
Regular action on 99 points
Generators in S99
(1 67 49)(2 68 50)(3 69 51)(4 70 52)(5 71 53)(6 72 54)(7 73 55)(8 74 56)(9 75 57)(10 76 58)(11 77 59)(12 78 60)(13 79 61)(14 80 62)(15 81 63)(16 82 64)(17 83 65)(18 84 66)(19 85 34)(20 86 35)(21 87 36)(22 88 37)(23 89 38)(24 90 39)(25 91 40)(26 92 41)(27 93 42)(28 94 43)(29 95 44)(30 96 45)(31 97 46)(32 98 47)(33 99 48)
(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)

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

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

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

C3×C33 is a maximal subgroup of   C3⋊D33

99 conjugacy classes

class 1 3A···3H11A···11J33A···33CB
order13···311···1133···33
size11···11···11···1

99 irreducible representations

dim1111
type+
imageC1C3C11C33
kernelC3×C33C33C32C3
# reps181080

Matrix representation of C3×C33 in GL2(𝔽67) generated by

370
029
,
650
017
G:=sub<GL(2,GF(67))| [37,0,0,29],[65,0,0,17] >;

C3×C33 in GAP, Magma, Sage, TeX

C_3\times C_{33}
% in TeX

G:=Group("C3xC33");
// GroupNames label

G:=SmallGroup(99,2);
// by ID

G=gap.SmallGroup(99,2);
# by ID

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

G:=Group<a,b|a^3=b^33=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C33 in TeX

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