metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C13⋊C9, C39.C3, C3.(C13⋊C3), SmallGroup(117,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊C9 |
Generators and relations for C13⋊C9
G = < a,b | a13=b9=1, bab-1=a9 >
Character table of C13⋊C9
| class | 1 | 3A | 3B | 9A | 9B | 9C | 9D | 9E | 9F | 13A | 13B | 13C | 13D | 39A | 39B | 39C | 39D | 39E | 39F | 39G | 39H | |
| size | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 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 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | ζ32 | ζ3 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ5 | 1 | ζ3 | ζ32 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ6 | 1 | ζ3 | ζ32 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ7 | 1 | ζ32 | ζ3 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ8 | 1 | ζ3 | ζ32 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ9 | 1 | ζ32 | ζ3 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ10 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | complex lifted from C13⋊C3 |
| ρ11 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | complex lifted from C13⋊C3 |
| ρ12 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | complex lifted from C13⋊C3 |
| ρ13 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | complex lifted from C13⋊C3 |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | complex faithful, Schur index 3 |
| ρ15 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | complex faithful, Schur index 3 |
| ρ16 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | complex faithful, Schur index 3 |
| ρ17 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | complex faithful, Schur index 3 |
| ρ18 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | complex faithful, Schur index 3 |
| ρ19 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | complex faithful, Schur index 3 |
| ρ20 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ138+ζ137 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | complex faithful, Schur index 3 |
| ρ21 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ133+ζ13 | ζ136+ζ135+ζ132 | ζ1312+ζ1310+ζ134 | ζ1311+ζ138+ζ137 | ζ32ζ139+ζ32ζ133+ζ32ζ13 | ζ32ζ136+ζ32ζ135+ζ32ζ132 | ζ3ζ1311+ζ3ζ138+ζ3ζ137 | ζ3ζ139+ζ3ζ133+ζ3ζ13 | ζ3ζ136+ζ3ζ135+ζ3ζ132 | ζ3ζ1312+ζ3ζ1310+ζ3ζ134 | ζ32ζ1312+ζ32ζ1310+ζ32ζ134 | ζ32ζ1311+ζ32ζ138+ζ32ζ137 | complex faithful, Schur index 3 |
►Regular action on 117 points
Generators in S117
(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)
(1 107 77 33 96 57 16 91 43)(2 110 73 34 99 53 17 81 52)(3 113 69 35 102 62 18 84 48)(4 116 78 36 92 58 19 87 44)(5 106 74 37 95 54 20 90 40)(6 109 70 38 98 63 21 80 49)(7 112 66 39 101 59 22 83 45)(8 115 75 27 104 55 23 86 41)(9 105 71 28 94 64 24 89 50)(10 108 67 29 97 60 25 79 46)(11 111 76 30 100 56 26 82 42)(12 114 72 31 103 65 14 85 51)(13 117 68 32 93 61 15 88 47)
G:=sub<Sym(117)| (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), (1,107,77,33,96,57,16,91,43)(2,110,73,34,99,53,17,81,52)(3,113,69,35,102,62,18,84,48)(4,116,78,36,92,58,19,87,44)(5,106,74,37,95,54,20,90,40)(6,109,70,38,98,63,21,80,49)(7,112,66,39,101,59,22,83,45)(8,115,75,27,104,55,23,86,41)(9,105,71,28,94,64,24,89,50)(10,108,67,29,97,60,25,79,46)(11,111,76,30,100,56,26,82,42)(12,114,72,31,103,65,14,85,51)(13,117,68,32,93,61,15,88,47)>;
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)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,107,77,33,96,57,16,91,43)(2,110,73,34,99,53,17,81,52)(3,113,69,35,102,62,18,84,48)(4,116,78,36,92,58,19,87,44)(5,106,74,37,95,54,20,90,40)(6,109,70,38,98,63,21,80,49)(7,112,66,39,101,59,22,83,45)(8,115,75,27,104,55,23,86,41)(9,105,71,28,94,64,24,89,50)(10,108,67,29,97,60,25,79,46)(11,111,76,30,100,56,26,82,42)(12,114,72,31,103,65,14,85,51)(13,117,68,32,93,61,15,88,47) );
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),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,107,77,33,96,57,16,91,43),(2,110,73,34,99,53,17,81,52),(3,113,69,35,102,62,18,84,48),(4,116,78,36,92,58,19,87,44),(5,106,74,37,95,54,20,90,40),(6,109,70,38,98,63,21,80,49),(7,112,66,39,101,59,22,83,45),(8,115,75,27,104,55,23,86,41),(9,105,71,28,94,64,24,89,50),(10,108,67,29,97,60,25,79,46),(11,111,76,30,100,56,26,82,42),(12,114,72,31,103,65,14,85,51),(13,117,68,32,93,61,15,88,47)]])
C13⋊C9 is a maximal subgroup of
C13⋊C18 C9×C13⋊C3 C117⋊C3 C117⋊3C3 C39.C32 C39.A4
C13⋊C9 is a maximal quotient of C13⋊C27 C39.A4
Matrix representation of C13⋊C9 ►in GL3(𝔽937) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 730 | 454 |
| 643 | 281 | 633 |
| 85 | 427 | 250 |
| 532 | 848 | 804 |
G:=sub<GL(3,GF(937))| [0,0,1,1,0,730,0,1,454],[643,85,532,281,427,848,633,250,804] >;
C13⋊C9 in GAP, Magma, Sage, TeX
C_{13}\rtimes C_9 % in TeX
G:=Group("C13:C9"); // GroupNames label
G:=SmallGroup(117,1);
// by ID
G=gap.SmallGroup(117,1);
# by ID
G:=PCGroup([3,-3,-3,-13,9,245]);
// Polycyclic
G:=Group<a,b|a^13=b^9=1,b*a*b^-1=a^9>;
// generators/relations
Export
Subgroup lattice of C13⋊C9 in TeX
Character table of C13⋊C9 in TeX