metacyclic, supersoluble, monomial
Aliases: C63⋊6C6, C7⋊4(C9⋊C6), D9⋊2(C7⋊C3), (C7×D9)⋊2C3, C63⋊C3⋊2C2, C21.7(C3×S3), C9⋊2(C2×C7⋊C3), C3.2(S3×C7⋊C3), (C3×C7⋊C3).4S3, SmallGroup(378,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — C63⋊6C6 |
Generators and relations for C63⋊6C6
G = < a,b | a63=b6=1, bab-1=a23 >
Character table of C63⋊6C6
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 7A | 7B | 9A | 9B | 9C | 14A | 14B | 21A | 21B | 63A | 63B | 63C | 63D | 63E | 63F | |
| size | 1 | 9 | 2 | 21 | 21 | 63 | 63 | 3 | 3 | 6 | 42 | 42 | 27 | 27 | 6 | 6 | 6 | 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 | trivial |
| ρ2 | 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 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | 2 | -1 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | 2 | -1 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ11 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ12 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 3 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ13 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 3 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ15 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | -3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ16 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | -3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | complex faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | complex faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | complex faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | complex faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | complex faithful |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ98ζ74-ζ98ζ7-ζ94ζ72+ζ94ζ7+ζ92ζ72-ζ92ζ7+ζ9ζ74-ζ9ζ72 | -ζ98ζ74+ζ98ζ72-ζ97ζ72+ζ97ζ7+ζ94ζ74-ζ94ζ72-ζ92ζ74+ζ92ζ7 | ζ95ζ76-ζ95ζ73+ζ94ζ76-ζ94ζ75+ζ92ζ75-ζ92ζ73-ζ9ζ75+ζ9ζ73 | -ζ95ζ75+ζ95ζ73-ζ94ζ76+ζ94ζ73+ζ92ζ76-ζ92ζ75-ζ9ζ76+ζ9ζ75 | -ζ97ζ75+ζ97ζ73-ζ95ζ76+ζ95ζ75-ζ92ζ76+ζ92ζ73+ζ9ζ76-ζ9ζ75 | ζ95ζ72-ζ95ζ7-ζ94ζ74+ζ94ζ72+ζ92ζ74-ζ92ζ7-ζ9ζ74+ζ9ζ7 | complex faithful |
►On 63 points
Generators in S63
(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)
(2 12 59 9 26 24)(3 23 54 17 51 47)(4 34 49 25 13 7)(5 45 44 33 38 30)(6 56 39 41 63 53)(8 15 29 57 50 36)(10 37 19)(11 48 14 18 62 42)(16 40 52 58 61 31)(20 21 32 27 35 60)(22 43)(28 46 55)
G:=sub<Sym(63)| (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), (2,12,59,9,26,24)(3,23,54,17,51,47)(4,34,49,25,13,7)(5,45,44,33,38,30)(6,56,39,41,63,53)(8,15,29,57,50,36)(10,37,19)(11,48,14,18,62,42)(16,40,52,58,61,31)(20,21,32,27,35,60)(22,43)(28,46,55)>;
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), (2,12,59,9,26,24)(3,23,54,17,51,47)(4,34,49,25,13,7)(5,45,44,33,38,30)(6,56,39,41,63,53)(8,15,29,57,50,36)(10,37,19)(11,48,14,18,62,42)(16,40,52,58,61,31)(20,21,32,27,35,60)(22,43)(28,46,55) );
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)], [(2,12,59,9,26,24),(3,23,54,17,51,47),(4,34,49,25,13,7),(5,45,44,33,38,30),(6,56,39,41,63,53),(8,15,29,57,50,36),(10,37,19),(11,48,14,18,62,42),(16,40,52,58,61,31),(20,21,32,27,35,60),(22,43),(28,46,55)]])
Matrix representation of C63⋊6C6 ►in GL6(𝔽127)
| 100 | 76 | 106 | 20 | 57 | 95 |
| 51 | 24 | 107 | 86 | 32 | 89 |
| 57 | 95 | 116 | 18 | 65 | 121 |
| 32 | 89 | 109 | 98 | 6 | 71 |
| 65 | 121 | 24 | 100 | 18 | 29 |
| 6 | 71 | 27 | 51 | 98 | 116 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 126 | 126 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 126 | 126 |
| 104 | 0 | 126 | 0 | 126 | 0 |
| 23 | 23 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(127))| [100,51,57,32,65,6,76,24,95,89,121,71,106,107,116,109,24,27,20,86,18,98,100,51,57,32,65,6,18,98,95,89,121,71,29,116],[1,126,0,0,104,23,0,126,0,0,0,23,0,0,0,0,126,1,0,0,0,0,0,1,0,0,1,126,126,1,0,0,0,126,0,1] >;
C63⋊6C6 in GAP, Magma, Sage, TeX
C_{63}\rtimes_6C_6 % in TeX
G:=Group("C63:6C6"); // GroupNames label
G:=SmallGroup(378,14);
// by ID
G=gap.SmallGroup(378,14);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,3962,997,327,368,6304]);
// Polycyclic
G:=Group<a,b|a^63=b^6=1,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C63⋊6C6 in TeX
Character table of C63⋊6C6 in TeX