metacyclic, supersoluble, monomial, Z-group
Aliases: C13⋊C18, D13⋊C9, C39.C6, C13⋊C9⋊C2, C3.(C13⋊C6), (C3×D13).C3, SmallGroup(234,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C18 |
Generators and relations for C13⋊C18
G = < a,b | a13=b18=1, bab-1=a10 >
Character table of C13⋊C18
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 13A | 13B | 18A | 18B | 18C | 18D | 18E | 18F | 39A | 39B | 39C | 39D | |
| size | 1 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ94 | ζ9 | ζ97 | ζ98 | ζ95 | ζ92 | 1 | 1 | -ζ97 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 18 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ98 | ζ92 | ζ95 | ζ97 | ζ9 | ζ94 | 1 | 1 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ9 | ζ97 | ζ94 | ζ92 | ζ98 | ζ95 | 1 | 1 | -ζ94 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 18 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ98 | ζ92 | ζ95 | ζ97 | ζ9 | ζ94 | 1 | 1 | -ζ95 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 18 |
| ρ11 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ94 | ζ9 | ζ97 | ζ98 | ζ95 | ζ92 | 1 | 1 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ9 | ζ97 | ζ94 | ζ92 | ζ98 | ζ95 | 1 | 1 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ13 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ95 | ζ98 | ζ92 | ζ9 | ζ94 | ζ97 | 1 | 1 | -ζ92 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 18 |
| ρ14 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ97 | ζ94 | ζ9 | ζ95 | ζ92 | ζ98 | 1 | 1 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ95 | ζ98 | ζ92 | ζ9 | ζ94 | ζ97 | 1 | 1 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ97 | ζ94 | ζ9 | ζ95 | ζ92 | ζ98 | 1 | 1 | -ζ9 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 18 |
| ρ17 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ92 | ζ95 | ζ98 | ζ94 | ζ97 | ζ9 | 1 | 1 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ18 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ92 | ζ95 | ζ98 | ζ94 | ζ97 | ζ9 | 1 | 1 | -ζ98 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 18 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ21 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | complex faithful, Schur index 3 |
| ρ22 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | complex faithful, Schur index 3 |
| ρ23 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | complex faithful, Schur index 3 |
| ρ24 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ1312+ζ3ζ1310+ζ3ζ139+ζ3ζ134+ζ3ζ133+ζ3ζ13 | ζ3ζ1311+ζ3ζ138+ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1311+ζ32ζ138+ζ32ζ137+ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ139+ζ32ζ134+ζ32ζ133+ζ32ζ13 | complex faithful, Schur index 3 |
►On 117 points
Generators in S117
(1 87 61 29 70 102 11 20 111 79 38 52 96)(2 39 21 71 88 53 112 103 62 97 80 12 30)(3 81 104 89 40 13 63 54 22 31 98 113 72)(4 99 55 41 64 114 23 14 105 73 32 46 90)(5 33 15 65 82 47 106 115 56 91 74 24 42)(6 75 116 83 34 25 57 48 16 43 92 107 66)(7 93 49 35 76 108 17 26 117 67 44 58 84)(8 45 27 77 94 59 100 109 50 85 68 18 36)(9 69 110 95 28 19 51 60 10 37 86 101 78)
(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)(100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117)
G:=sub<Sym(117)| (1,87,61,29,70,102,11,20,111,79,38,52,96)(2,39,21,71,88,53,112,103,62,97,80,12,30)(3,81,104,89,40,13,63,54,22,31,98,113,72)(4,99,55,41,64,114,23,14,105,73,32,46,90)(5,33,15,65,82,47,106,115,56,91,74,24,42)(6,75,116,83,34,25,57,48,16,43,92,107,66)(7,93,49,35,76,108,17,26,117,67,44,58,84)(8,45,27,77,94,59,100,109,50,85,68,18,36)(9,69,110,95,28,19,51,60,10,37,86,101,78), (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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117)>;
G:=Group( (1,87,61,29,70,102,11,20,111,79,38,52,96)(2,39,21,71,88,53,112,103,62,97,80,12,30)(3,81,104,89,40,13,63,54,22,31,98,113,72)(4,99,55,41,64,114,23,14,105,73,32,46,90)(5,33,15,65,82,47,106,115,56,91,74,24,42)(6,75,116,83,34,25,57,48,16,43,92,107,66)(7,93,49,35,76,108,17,26,117,67,44,58,84)(8,45,27,77,94,59,100,109,50,85,68,18,36)(9,69,110,95,28,19,51,60,10,37,86,101,78), (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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117) );
G=PermutationGroup([[(1,87,61,29,70,102,11,20,111,79,38,52,96),(2,39,21,71,88,53,112,103,62,97,80,12,30),(3,81,104,89,40,13,63,54,22,31,98,113,72),(4,99,55,41,64,114,23,14,105,73,32,46,90),(5,33,15,65,82,47,106,115,56,91,74,24,42),(6,75,116,83,34,25,57,48,16,43,92,107,66),(7,93,49,35,76,108,17,26,117,67,44,58,84),(8,45,27,77,94,59,100,109,50,85,68,18,36),(9,69,110,95,28,19,51,60,10,37,86,101,78)], [(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),(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117)]])
C13⋊C18 is a maximal subgroup of
C13⋊C36
C13⋊C18 is a maximal quotient of C13⋊2C36
Matrix representation of C13⋊C18 ►in GL6(𝔽937)
| 936 | 1 | 0 | 0 | 0 | 0 |
| 936 | 0 | 1 | 0 | 0 | 0 |
| 936 | 0 | 0 | 1 | 0 | 0 |
| 936 | 0 | 0 | 0 | 1 | 0 |
| 936 | 0 | 0 | 0 | 0 | 1 |
| 274 | 663 | 273 | 664 | 274 | 662 |
| 675 | 868 | 532 | 596 | 501 | 708 |
| 661 | 742 | 366 | 571 | 195 | 276 |
| 561 | 619 | 591 | 729 | 834 | 516 |
| 624 | 518 | 692 | 666 | 674 | 516 |
| 296 | 223 | 885 | 936 | 366 | 276 |
| 627 | 515 | 885 | 644 | 35 | 708 |
G:=sub<GL(6,GF(937))| [936,936,936,936,936,274,1,0,0,0,0,663,0,1,0,0,0,273,0,0,1,0,0,664,0,0,0,1,0,274,0,0,0,0,1,662],[675,661,561,624,296,627,868,742,619,518,223,515,532,366,591,692,885,885,596,571,729,666,936,644,501,195,834,674,366,35,708,276,516,516,276,708] >;
C13⋊C18 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_{18} % in TeX
G:=Group("C13:C18"); // GroupNames label
G:=SmallGroup(234,1);
// by ID
G=gap.SmallGroup(234,1);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,29,3459,439]);
// Polycyclic
G:=Group<a,b|a^13=b^18=1,b*a*b^-1=a^10>;
// generators/relations
Export
Subgroup lattice of C13⋊C18 in TeX
Character table of C13⋊C18 in TeX