metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C109⋊C3, SmallGroup(327,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C109 — C109⋊C3 |
Generators and relations for C109⋊C3
G = < a,b | a109=b3=1, bab-1=a45 >
Smallest permutation representation of C109⋊C3
►On 109 points: primitive
Generators in S109
(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 99 100 101 102 103 104 105 106 107 108 109)
(2 64 46)(3 18 91)(4 81 27)(5 35 72)(6 98 8)(7 52 53)(9 69 34)(10 23 79)(11 86 15)(12 40 60)(13 103 105)(14 57 41)(16 74 22)(17 28 67)(19 45 48)(20 108 93)(21 62 29)(24 33 55)(25 96 100)(26 50 36)(30 84 107)(31 38 43)(32 101 88)(37 89 95)(39 106 76)(42 77 102)(44 94 83)(47 65 109)(49 82 90)(51 99 71)(54 70 97)(56 87 78)(58 104 59)(61 75 85)(63 92 66)(68 80 73)
G:=sub<Sym(109)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109), (2,64,46)(3,18,91)(4,81,27)(5,35,72)(6,98,8)(7,52,53)(9,69,34)(10,23,79)(11,86,15)(12,40,60)(13,103,105)(14,57,41)(16,74,22)(17,28,67)(19,45,48)(20,108,93)(21,62,29)(24,33,55)(25,96,100)(26,50,36)(30,84,107)(31,38,43)(32,101,88)(37,89,95)(39,106,76)(42,77,102)(44,94,83)(47,65,109)(49,82,90)(51,99,71)(54,70,97)(56,87,78)(58,104,59)(61,75,85)(63,92,66)(68,80,73)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109), (2,64,46)(3,18,91)(4,81,27)(5,35,72)(6,98,8)(7,52,53)(9,69,34)(10,23,79)(11,86,15)(12,40,60)(13,103,105)(14,57,41)(16,74,22)(17,28,67)(19,45,48)(20,108,93)(21,62,29)(24,33,55)(25,96,100)(26,50,36)(30,84,107)(31,38,43)(32,101,88)(37,89,95)(39,106,76)(42,77,102)(44,94,83)(47,65,109)(49,82,90)(51,99,71)(54,70,97)(56,87,78)(58,104,59)(61,75,85)(63,92,66)(68,80,73) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109)], [(2,64,46),(3,18,91),(4,81,27),(5,35,72),(6,98,8),(7,52,53),(9,69,34),(10,23,79),(11,86,15),(12,40,60),(13,103,105),(14,57,41),(16,74,22),(17,28,67),(19,45,48),(20,108,93),(21,62,29),(24,33,55),(25,96,100),(26,50,36),(30,84,107),(31,38,43),(32,101,88),(37,89,95),(39,106,76),(42,77,102),(44,94,83),(47,65,109),(49,82,90),(51,99,71),(54,70,97),(56,87,78),(58,104,59),(61,75,85),(63,92,66),(68,80,73)]])
39 conjugacy classes
| class | 1 | 3A | 3B | 109A | ··· | 109AJ |
| order | 1 | 3 | 3 | 109 | ··· | 109 |
| size | 1 | 109 | 109 | 3 | ··· | 3 |
39 irreducible representations
| dim | 1 | 1 | 3 |
| type | + | ||
| image | C1 | C3 | C109⋊C3 |
| kernel | C109⋊C3 | C109 | C1 |
| # reps | 1 | 2 | 36 |
Matrix representation of C109⋊C3 ►in GL3(𝔽2617) generated by
| 1129 | 1 | 0 |
| 956 | 0 | 1 |
| 1498 | 1497 | 1650 |
| 2006 | 704 | 986 |
| 204 | 2597 | 2292 |
| 1768 | 28 | 631 |
G:=sub<GL(3,GF(2617))| [1129,956,1498,1,0,1497,0,1,1650],[2006,204,1768,704,2597,28,986,2292,631] >;
C109⋊C3 in GAP, Magma, Sage, TeX
C_{109}\rtimes C_3 % in TeX
G:=Group("C109:C3"); // GroupNames label
G:=SmallGroup(327,1);
// by ID
G=gap.SmallGroup(327,1);
# by ID
G:=PCGroup([2,-3,-109,757]);
// Polycyclic
G:=Group<a,b|a^109=b^3=1,b*a*b^-1=a^45>;
// generators/relations
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