direct product, abelian, monomial, 3-elementary
Aliases: C3×C33, SmallGroup(99,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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 61)(2 68 62)(3 69 63)(4 70 64)(5 71 65)(6 72 66)(7 73 34)(8 74 35)(9 75 36)(10 76 37)(11 77 38)(12 78 39)(13 79 40)(14 80 41)(15 81 42)(16 82 43)(17 83 44)(18 84 45)(19 85 46)(20 86 47)(21 87 48)(22 88 49)(23 89 50)(24 90 51)(25 91 52)(26 92 53)(27 93 54)(28 94 55)(29 95 56)(30 96 57)(31 97 58)(32 98 59)(33 99 60)
(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)| (1,67,61)(2,68,62)(3,69,63)(4,70,64)(5,71,65)(6,72,66)(7,73,34)(8,74,35)(9,75,36)(10,76,37)(11,77,38)(12,78,39)(13,79,40)(14,80,41)(15,81,42)(16,82,43)(17,83,44)(18,84,45)(19,85,46)(20,86,47)(21,87,48)(22,88,49)(23,89,50)(24,90,51)(25,91,52)(26,92,53)(27,93,54)(28,94,55)(29,95,56)(30,96,57)(31,97,58)(32,98,59)(33,99,60), (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( (1,67,61)(2,68,62)(3,69,63)(4,70,64)(5,71,65)(6,72,66)(7,73,34)(8,74,35)(9,75,36)(10,76,37)(11,77,38)(12,78,39)(13,79,40)(14,80,41)(15,81,42)(16,82,43)(17,83,44)(18,84,45)(19,85,46)(20,86,47)(21,87,48)(22,88,49)(23,89,50)(24,90,51)(25,91,52)(26,92,53)(27,93,54)(28,94,55)(29,95,56)(30,96,57)(31,97,58)(32,98,59)(33,99,60), (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([[(1,67,61),(2,68,62),(3,69,63),(4,70,64),(5,71,65),(6,72,66),(7,73,34),(8,74,35),(9,75,36),(10,76,37),(11,77,38),(12,78,39),(13,79,40),(14,80,41),(15,81,42),(16,82,43),(17,83,44),(18,84,45),(19,85,46),(20,86,47),(21,87,48),(22,88,49),(23,89,50),(24,90,51),(25,91,52),(26,92,53),(27,93,54),(28,94,55),(29,95,56),(30,96,57),(31,97,58),(32,98,59),(33,99,60)], [(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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