metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C103⋊C3, SmallGroup(309,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C103 — C103⋊C3 |
Generators and relations for C103⋊C3
G = < a,b | a103=b3=1, bab-1=a46 >
Smallest permutation representation of C103⋊C3
►On 103 points: primitive
Generators in S103
(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)
(2 57 47)(3 10 93)(4 66 36)(5 19 82)(6 75 25)(7 28 71)(8 84 14)(9 37 60)(11 46 49)(12 102 95)(13 55 38)(15 64 27)(16 17 73)(18 26 62)(20 35 51)(21 91 97)(22 44 40)(23 100 86)(24 53 29)(30 80 99)(31 33 42)(32 89 88)(34 98 77)(39 69 101)(41 78 90)(43 87 79)(45 96 68)(48 58 103)(50 67 92)(52 76 81)(54 85 70)(56 94 59)(61 65 83)(63 74 72)
G:=sub<Sym(103)| (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), (2,57,47)(3,10,93)(4,66,36)(5,19,82)(6,75,25)(7,28,71)(8,84,14)(9,37,60)(11,46,49)(12,102,95)(13,55,38)(15,64,27)(16,17,73)(18,26,62)(20,35,51)(21,91,97)(22,44,40)(23,100,86)(24,53,29)(30,80,99)(31,33,42)(32,89,88)(34,98,77)(39,69,101)(41,78,90)(43,87,79)(45,96,68)(48,58,103)(50,67,92)(52,76,81)(54,85,70)(56,94,59)(61,65,83)(63,74,72)>;
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), (2,57,47)(3,10,93)(4,66,36)(5,19,82)(6,75,25)(7,28,71)(8,84,14)(9,37,60)(11,46,49)(12,102,95)(13,55,38)(15,64,27)(16,17,73)(18,26,62)(20,35,51)(21,91,97)(22,44,40)(23,100,86)(24,53,29)(30,80,99)(31,33,42)(32,89,88)(34,98,77)(39,69,101)(41,78,90)(43,87,79)(45,96,68)(48,58,103)(50,67,92)(52,76,81)(54,85,70)(56,94,59)(61,65,83)(63,74,72) );
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)], [(2,57,47),(3,10,93),(4,66,36),(5,19,82),(6,75,25),(7,28,71),(8,84,14),(9,37,60),(11,46,49),(12,102,95),(13,55,38),(15,64,27),(16,17,73),(18,26,62),(20,35,51),(21,91,97),(22,44,40),(23,100,86),(24,53,29),(30,80,99),(31,33,42),(32,89,88),(34,98,77),(39,69,101),(41,78,90),(43,87,79),(45,96,68),(48,58,103),(50,67,92),(52,76,81),(54,85,70),(56,94,59),(61,65,83),(63,74,72)]])
37 conjugacy classes
| class | 1 | 3A | 3B | 103A | ··· | 103AH |
| order | 1 | 3 | 3 | 103 | ··· | 103 |
| size | 1 | 103 | 103 | 3 | ··· | 3 |
37 irreducible representations
| dim | 1 | 1 | 3 |
| type | + | ||
| image | C1 | C3 | C103⋊C3 |
| kernel | C103⋊C3 | C103 | C1 |
| # reps | 1 | 2 | 34 |
Matrix representation of C103⋊C3 ►in GL3(𝔽619) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 73 | 184 |
| 1 | 0 | 0 |
| 332 | 371 | 426 |
| 73 | 497 | 247 |
G:=sub<GL(3,GF(619))| [0,0,1,1,0,73,0,1,184],[1,332,73,0,371,497,0,426,247] >;
C103⋊C3 in GAP, Magma, Sage, TeX
C_{103}\rtimes C_3 % in TeX
G:=Group("C103:C3"); // GroupNames label
G:=SmallGroup(309,1);
// by ID
G=gap.SmallGroup(309,1);
# by ID
G:=PCGroup([2,-3,-103,673]);
// Polycyclic
G:=Group<a,b|a^103=b^3=1,b*a*b^-1=a^46>;
// generators/relations
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