direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C3×C31⋊C5, C93⋊C5, C31⋊2C15, SmallGroup(465,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C3×C31⋊C5 |
Generators and relations for C3×C31⋊C5
G = < a,b,c | a3=b31=c5=1, ab=ba, ac=ca, cbc-1=b2 >
Smallest permutation representation of C3×C31⋊C5
►On 93 points
Generators in S93
(1 63 32)(2 64 33)(3 65 34)(4 66 35)(5 67 36)(6 68 37)(7 69 38)(8 70 39)(9 71 40)(10 72 41)(11 73 42)(12 74 43)(13 75 44)(14 76 45)(15 77 46)(16 78 47)(17 79 48)(18 80 49)(19 81 50)(20 82 51)(21 83 52)(22 84 53)(23 85 54)(24 86 55)(25 87 56)(26 88 57)(27 89 58)(28 90 59)(29 91 60)(30 92 61)(31 93 62)
(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)
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(33 48 40 36 34)(35 49 56 44 38)(37 50 41 52 42)(39 51 57 60 46)(43 53 58 45 54)(47 55 59 61 62)(64 79 71 67 65)(66 80 87 75 69)(68 81 72 83 73)(70 82 88 91 77)(74 84 89 76 85)(78 86 90 92 93)
G:=sub<Sym(93)| (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31)(33,48,40,36,34)(35,49,56,44,38)(37,50,41,52,42)(39,51,57,60,46)(43,53,58,45,54)(47,55,59,61,62)(64,79,71,67,65)(66,80,87,75,69)(68,81,72,83,73)(70,82,88,91,77)(74,84,89,76,85)(78,86,90,92,93)>;
G:=Group( (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31)(33,48,40,36,34)(35,49,56,44,38)(37,50,41,52,42)(39,51,57,60,46)(43,53,58,45,54)(47,55,59,61,62)(64,79,71,67,65)(66,80,87,75,69)(68,81,72,83,73)(70,82,88,91,77)(74,84,89,76,85)(78,86,90,92,93) );
G=PermutationGroup([[(1,63,32),(2,64,33),(3,65,34),(4,66,35),(5,67,36),(6,68,37),(7,69,38),(8,70,39),(9,71,40),(10,72,41),(11,73,42),(12,74,43),(13,75,44),(14,76,45),(15,77,46),(16,78,47),(17,79,48),(18,80,49),(19,81,50),(20,82,51),(21,83,52),(22,84,53),(23,85,54),(24,86,55),(25,87,56),(26,88,57),(27,89,58),(28,90,59),(29,91,60),(30,92,61),(31,93,62)], [(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)], [(2,17,9,5,3),(4,18,25,13,7),(6,19,10,21,11),(8,20,26,29,15),(12,22,27,14,23),(16,24,28,30,31),(33,48,40,36,34),(35,49,56,44,38),(37,50,41,52,42),(39,51,57,60,46),(43,53,58,45,54),(47,55,59,61,62),(64,79,71,67,65),(66,80,87,75,69),(68,81,72,83,73),(70,82,88,91,77),(74,84,89,76,85),(78,86,90,92,93)]])
33 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 15A | ··· | 15H | 31A | ··· | 31F | 93A | ··· | 93L |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 31 | ··· | 31 | 93 | ··· | 93 |
| size | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 31 | ··· | 31 | 5 | ··· | 5 | 5 | ··· | 5 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 5 | 5 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C31⋊C5 | C3×C31⋊C5 |
| kernel | C3×C31⋊C5 | C31⋊C5 | C93 | C31 | C3 | C1 |
| # reps | 1 | 2 | 4 | 8 | 6 | 12 |
Matrix representation of C3×C31⋊C5 ►in GL5(𝔽1861)
| 454 | 0 | 0 | 0 | 0 |
| 0 | 454 | 0 | 0 | 0 |
| 0 | 0 | 454 | 0 | 0 |
| 0 | 0 | 0 | 454 | 0 |
| 0 | 0 | 0 | 0 | 454 |
| 1591 | 635 | 1612 | 708 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1153 | 524 | 1349 | 1729 | 904 |
| 20 | 721 | 1769 | 930 | 131 |
G:=sub<GL(5,GF(1861))| [454,0,0,0,0,0,454,0,0,0,0,0,454,0,0,0,0,0,454,0,0,0,0,0,454],[1591,1,0,0,0,635,0,1,0,0,1612,0,0,1,0,708,0,0,0,1,1,0,0,0,0],[1,0,0,1153,20,0,0,0,524,721,0,1,0,1349,1769,0,0,0,1729,930,0,0,1,904,131] >;
C3×C31⋊C5 in GAP, Magma, Sage, TeX
C_3\times C_{31}\rtimes C_5 % in TeX
G:=Group("C3xC31:C5"); // GroupNames label
G:=SmallGroup(465,2);
// by ID
G=gap.SmallGroup(465,2);
# by ID
G:=PCGroup([3,-3,-5,-31,725]);
// Polycyclic
G:=Group<a,b,c|a^3=b^31=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations
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