direct product, abelian, monomial, 3-elementary
Aliases: C3×C36, SmallGroup(108,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C36 |
| C1 — C3×C36 |
| C1 — C3×C36 |
Generators and relations for C3×C36
G = < a,b | a3=b36=1, ab=ba >
Smallest permutation representation of C3×C36
►Regular action on 108 points
Generators in S108
(1 108 39)(2 73 40)(3 74 41)(4 75 42)(5 76 43)(6 77 44)(7 78 45)(8 79 46)(9 80 47)(10 81 48)(11 82 49)(12 83 50)(13 84 51)(14 85 52)(15 86 53)(16 87 54)(17 88 55)(18 89 56)(19 90 57)(20 91 58)(21 92 59)(22 93 60)(23 94 61)(24 95 62)(25 96 63)(26 97 64)(27 98 65)(28 99 66)(29 100 67)(30 101 68)(31 102 69)(32 103 70)(33 104 71)(34 105 72)(35 106 37)(36 107 38)
(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 99 100 101 102 103 104 105 106 107 108)
G:=sub<Sym(108)| (1,108,39)(2,73,40)(3,74,41)(4,75,42)(5,76,43)(6,77,44)(7,78,45)(8,79,46)(9,80,47)(10,81,48)(11,82,49)(12,83,50)(13,84,51)(14,85,52)(15,86,53)(16,87,54)(17,88,55)(18,89,56)(19,90,57)(20,91,58)(21,92,59)(22,93,60)(23,94,61)(24,95,62)(25,96,63)(26,97,64)(27,98,65)(28,99,66)(29,100,67)(30,101,68)(31,102,69)(32,103,70)(33,104,71)(34,105,72)(35,106,37)(36,107,38), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)>;
G:=Group( (1,108,39)(2,73,40)(3,74,41)(4,75,42)(5,76,43)(6,77,44)(7,78,45)(8,79,46)(9,80,47)(10,81,48)(11,82,49)(12,83,50)(13,84,51)(14,85,52)(15,86,53)(16,87,54)(17,88,55)(18,89,56)(19,90,57)(20,91,58)(21,92,59)(22,93,60)(23,94,61)(24,95,62)(25,96,63)(26,97,64)(27,98,65)(28,99,66)(29,100,67)(30,101,68)(31,102,69)(32,103,70)(33,104,71)(34,105,72)(35,106,37)(36,107,38), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108) );
G=PermutationGroup([[(1,108,39),(2,73,40),(3,74,41),(4,75,42),(5,76,43),(6,77,44),(7,78,45),(8,79,46),(9,80,47),(10,81,48),(11,82,49),(12,83,50),(13,84,51),(14,85,52),(15,86,53),(16,87,54),(17,88,55),(18,89,56),(19,90,57),(20,91,58),(21,92,59),(22,93,60),(23,94,61),(24,95,62),(25,96,63),(26,97,64),(27,98,65),(28,99,66),(29,100,67),(30,101,68),(31,102,69),(32,103,70),(33,104,71),(34,105,72),(35,106,37),(36,107,38)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)]])
C3×C36 is a maximal subgroup of
C36.S3 C12.D9 C36⋊S3
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 9A | ··· | 9R | 12A | ··· | 12P | 18A | ··· | 18R | 36A | ··· | 36AJ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C36 |
| kernel | C3×C36 | C3×C18 | C36 | C3×C12 | C3×C9 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 18 | 12 | 4 | 18 | 36 |
Matrix representation of C3×C36 ►in GL2(𝔽37) generated by
| 10 | 0 |
| 0 | 1 |
| 3 | 0 |
| 0 | 20 |
G:=sub<GL(2,GF(37))| [10,0,0,1],[3,0,0,20] >;
C3×C36 in GAP, Magma, Sage, TeX
C_3\times C_{36} % in TeX
G:=Group("C3xC36"); // GroupNames label
G:=SmallGroup(108,12);
// by ID
G=gap.SmallGroup(108,12);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-3,90,186]);
// Polycyclic
G:=Group<a,b|a^3=b^36=1,a*b=b*a>;
// generators/relations
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