metacyclic, supersoluble, monomial, A-group
Aliases: C6.F7, C21⋊1C12, C42.1C6, Dic21⋊C3, C3⋊(C7⋊C12), C7⋊C3⋊Dic3, C7⋊(C3×Dic3), C14.(C3×S3), C2.(C3⋊F7), (C3×C7⋊C3)⋊1C4, (C2×C7⋊C3).S3, (C6×C7⋊C3).1C2, SmallGroup(252,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C6.F7 |
Generators and relations for C6.F7
G = < a,b,c | a6=b7=1, c6=a3, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C6.F7
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 7 | 12A | 12B | 12C | 12D | 14 | 21A | 21B | 42A | 42B | |
| size | 1 | 1 | 2 | 7 | 7 | 14 | 14 | 21 | 21 | 2 | 7 | 7 | 14 | 14 | 6 | 21 | 21 | 21 | 21 | 6 | 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 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -i | i | -1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | -1 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | i | -i | -1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | -1 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -i | i | -1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | -1 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | i | -i | -1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | -1 | 1 | 1 | -1 | -1 | linear of order 12 |
| ρ13 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 2 | 0 | 0 | 0 | 0 | -2 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 1 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | 2 | 0 | 0 | 0 | 0 | -2 | -1 | -1 | 1 | 1 | complex lifted from C3×Dic3 |
| ρ16 | 2 | 2 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ17 | 2 | 2 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ18 | 2 | -2 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 1 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | 2 | 0 | 0 | 0 | 0 | -2 | -1 | -1 | 1 | 1 | complex lifted from C3×Dic3 |
| ρ19 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ20 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | 1-√21/2 | orthogonal lifted from C3⋊F7 |
| ρ21 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | 1+√21/2 | orthogonal lifted from C3⋊F7 |
| ρ22 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1-√21/2 | 1+√21/2 | -1-√21/2 | -1+√21/2 | symplectic faithful, Schur index 2 |
| ρ23 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1+√21/2 | 1-√21/2 | -1+√21/2 | -1-√21/2 | symplectic faithful, Schur index 2 |
| ρ24 | 6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from C7⋊C12, Schur index 2 |
►On 84 points
Generators in S84
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 78 28 19 84 34)(14 35 73 20 29 79)(15 80 30 21 74 36)(16 25 75 22 31 81)(17 82 32 23 76 26)(18 27 77 24 33 83)(37 52 63 43 58 69)(38 70 59 44 64 53)(39 54 65 45 60 71)(40 72 49 46 66 55)(41 56 67 47 50 61)(42 62 51 48 68 57)
(1 67 58 17 45 80 28)(2 81 18 68 29 46 59)(3 47 69 82 60 30 19)(4 31 83 48 20 49 70)(5 50 37 32 71 21 84)(6 22 33 51 73 72 38)(7 61 52 23 39 74 34)(8 75 24 62 35 40 53)(9 41 63 76 54 36 13)(10 25 77 42 14 55 64)(11 56 43 26 65 15 78)(12 16 27 57 79 66 44)
(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)
G:=sub<Sym(84)| (1,3,5,7,9,11)(2,12,10,8,6,4)(13,78,28,19,84,34)(14,35,73,20,29,79)(15,80,30,21,74,36)(16,25,75,22,31,81)(17,82,32,23,76,26)(18,27,77,24,33,83)(37,52,63,43,58,69)(38,70,59,44,64,53)(39,54,65,45,60,71)(40,72,49,46,66,55)(41,56,67,47,50,61)(42,62,51,48,68,57), (1,67,58,17,45,80,28)(2,81,18,68,29,46,59)(3,47,69,82,60,30,19)(4,31,83,48,20,49,70)(5,50,37,32,71,21,84)(6,22,33,51,73,72,38)(7,61,52,23,39,74,34)(8,75,24,62,35,40,53)(9,41,63,76,54,36,13)(10,25,77,42,14,55,64)(11,56,43,26,65,15,78)(12,16,27,57,79,66,44), (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)>;
G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4)(13,78,28,19,84,34)(14,35,73,20,29,79)(15,80,30,21,74,36)(16,25,75,22,31,81)(17,82,32,23,76,26)(18,27,77,24,33,83)(37,52,63,43,58,69)(38,70,59,44,64,53)(39,54,65,45,60,71)(40,72,49,46,66,55)(41,56,67,47,50,61)(42,62,51,48,68,57), (1,67,58,17,45,80,28)(2,81,18,68,29,46,59)(3,47,69,82,60,30,19)(4,31,83,48,20,49,70)(5,50,37,32,71,21,84)(6,22,33,51,73,72,38)(7,61,52,23,39,74,34)(8,75,24,62,35,40,53)(9,41,63,76,54,36,13)(10,25,77,42,14,55,64)(11,56,43,26,65,15,78)(12,16,27,57,79,66,44), (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) );
G=PermutationGroup([[(1,3,5,7,9,11),(2,12,10,8,6,4),(13,78,28,19,84,34),(14,35,73,20,29,79),(15,80,30,21,74,36),(16,25,75,22,31,81),(17,82,32,23,76,26),(18,27,77,24,33,83),(37,52,63,43,58,69),(38,70,59,44,64,53),(39,54,65,45,60,71),(40,72,49,46,66,55),(41,56,67,47,50,61),(42,62,51,48,68,57)], [(1,67,58,17,45,80,28),(2,81,18,68,29,46,59),(3,47,69,82,60,30,19),(4,31,83,48,20,49,70),(5,50,37,32,71,21,84),(6,22,33,51,73,72,38),(7,61,52,23,39,74,34),(8,75,24,62,35,40,53),(9,41,63,76,54,36,13),(10,25,77,42,14,55,64),(11,56,43,26,65,15,78),(12,16,27,57,79,66,44)], [(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)]])
Matrix representation of C6.F7 ►in GL6(𝔽337)
| 292 | 246 | 246 | 0 | 246 | 0 |
| 0 | 292 | 246 | 246 | 0 | 246 |
| 91 | 91 | 46 | 0 | 0 | 91 |
| 246 | 0 | 0 | 292 | 246 | 246 |
| 91 | 0 | 91 | 91 | 46 | 0 |
| 0 | 91 | 0 | 91 | 91 | 46 |
| 336 | 336 | 336 | 336 | 336 | 336 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 330 | 14 | 81 | 148 | 221 | 168 |
| 67 | 134 | 207 | 154 | 323 | 316 |
| 73 | 20 | 189 | 182 | 203 | 270 |
| 169 | 162 | 183 | 250 | 317 | 53 |
| 21 | 88 | 155 | 228 | 175 | 7 |
| 67 | 140 | 87 | 256 | 249 | 270 |
G:=sub<GL(6,GF(337))| [292,0,91,246,91,0,246,292,91,0,0,91,246,246,46,0,91,0,0,246,0,292,91,91,246,0,0,246,46,91,0,246,91,246,0,46],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[330,67,73,169,21,67,14,134,20,162,88,140,81,207,189,183,155,87,148,154,182,250,228,256,221,323,203,317,175,249,168,316,270,53,7,270] >;
C6.F7 in GAP, Magma, Sage, TeX
C_6.F_7
% in TeX
G:=Group("C6.F7"); // GroupNames label
G:=SmallGroup(252,18);
// by ID
G=gap.SmallGroup(252,18);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404,909]);
// Polycyclic
G:=Group<a,b,c|a^6=b^7=1,c^6=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C6.F7 in TeX
Character table of C6.F7 in TeX