direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7×C13⋊C3, C13⋊C21, C91⋊2C3, SmallGroup(273,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C7×C13⋊C3 |
Generators and relations for C7×C13⋊C3
G = < a,b,c | a7=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C7×C13⋊C3
►On 91 points
Generators in S91
(1 79 66 53 40 27 14)(2 80 67 54 41 28 15)(3 81 68 55 42 29 16)(4 82 69 56 43 30 17)(5 83 70 57 44 31 18)(6 84 71 58 45 32 19)(7 85 72 59 46 33 20)(8 86 73 60 47 34 21)(9 87 74 61 48 35 22)(10 88 75 62 49 36 23)(11 89 76 63 50 37 24)(12 90 77 64 51 38 25)(13 91 78 65 52 39 26)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)
G:=sub<Sym(91)| (1,79,66,53,40,27,14)(2,80,67,54,41,28,15)(3,81,68,55,42,29,16)(4,82,69,56,43,30,17)(5,83,70,57,44,31,18)(6,84,71,58,45,32,19)(7,85,72,59,46,33,20)(8,86,73,60,47,34,21)(9,87,74,61,48,35,22)(10,88,75,62,49,36,23)(11,89,76,63,50,37,24)(12,90,77,64,51,38,25)(13,91,78,65,52,39,26), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)>;
G:=Group( (1,79,66,53,40,27,14)(2,80,67,54,41,28,15)(3,81,68,55,42,29,16)(4,82,69,56,43,30,17)(5,83,70,57,44,31,18)(6,84,71,58,45,32,19)(7,85,72,59,46,33,20)(8,86,73,60,47,34,21)(9,87,74,61,48,35,22)(10,88,75,62,49,36,23)(11,89,76,63,50,37,24)(12,90,77,64,51,38,25)(13,91,78,65,52,39,26), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90) );
G=PermutationGroup([[(1,79,66,53,40,27,14),(2,80,67,54,41,28,15),(3,81,68,55,42,29,16),(4,82,69,56,43,30,17),(5,83,70,57,44,31,18),(6,84,71,58,45,32,19),(7,85,72,59,46,33,20),(8,86,73,60,47,34,21),(9,87,74,61,48,35,22),(10,88,75,62,49,36,23),(11,89,76,63,50,37,24),(12,90,77,64,51,38,25),(13,91,78,65,52,39,26)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(67,69,75),(68,72,71),(70,78,76),(73,74,77),(80,82,88),(81,85,84),(83,91,89),(86,87,90)]])
49 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7F | 13A | 13B | 13C | 13D | 21A | ··· | 21L | 91A | ··· | 91X |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 13 | 13 | 13 | 13 | 21 | ··· | 21 | 91 | ··· | 91 |
| size | 1 | 13 | 13 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | ··· | 3 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C7 | C21 | C13⋊C3 | C7×C13⋊C3 |
| kernel | C7×C13⋊C3 | C91 | C13⋊C3 | C13 | C7 | C1 |
| # reps | 1 | 2 | 6 | 12 | 4 | 24 |
Matrix representation of C7×C13⋊C3 ►in GL3(𝔽547) generated by
| 520 | 0 | 0 |
| 0 | 520 | 0 |
| 0 | 0 | 520 |
| 253 | 464 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 82 | 252 | 464 |
| 72 | 465 | 294 |
G:=sub<GL(3,GF(547))| [520,0,0,0,520,0,0,0,520],[253,1,0,464,0,1,1,0,0],[1,82,72,0,252,465,0,464,294] >;
C7×C13⋊C3 in GAP, Magma, Sage, TeX
C_7\times C_{13}\rtimes C_3 % in TeX
G:=Group("C7xC13:C3"); // GroupNames label
G:=SmallGroup(273,2);
// by ID
G=gap.SmallGroup(273,2);
# by ID
G:=PCGroup([3,-3,-7,-13,569]);
// Polycyclic
G:=Group<a,b,c|a^7=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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