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G = C3×C33order 99 = 32·11

Abelian group of type [3,33]

Aliases: C3×C33, SmallGroup(99,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C33
 Chief series C1 — C11 — C33 — C3×C33
 Lower central C1 — C3×C33
 Upper central 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 ··· 3H 11A ··· 11J 33A ··· 33CB order 1 3 ··· 3 11 ··· 11 33 ··· 33 size 1 1 ··· 1 1 ··· 1 1 ··· 1

99 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C11 C33 kernel C3×C33 C33 C32 C3 # reps 1 8 10 80

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

 37 0 0 29
,
 65 0 0 17
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

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