metacyclic, supersoluble, monomial
Aliases: C63⋊5C6, C7⋊3(C9⋊C6), D9⋊1(C7⋊C3), (C7×D9)⋊1C3, C63⋊3C3⋊2C2, C21.6(C3×S3), C9⋊1(C2×C7⋊C3), C3.1(S3×C7⋊C3), (C3×C7⋊C3).3S3, SmallGroup(378,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — C63⋊C6 |
Generators and relations for C63⋊C6
G = < a,b | a63=b6=1, bab-1=a2 >
Character table of C63⋊C6
| 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 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | ζ98ζ74-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-ζ9ζ74+ζ9ζ72 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | 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 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-ζ9ζ74+ζ9ζ72 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | 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 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-ζ9ζ74+ζ9ζ72 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | 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 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-ζ92ζ7-ζ9ζ74+ζ9ζ7 | ζ98ζ74-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-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 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-ζ9ζ74+ζ9ζ72 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-ζ92ζ7-ζ9ζ74+ζ9ζ7 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | ζ98ζ74-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | 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-ζ98ζ72+ζ94ζ72-ζ94ζ7-ζ92ζ72+ζ92ζ7+ζ9ζ74-ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ74+ζ94ζ7+ζ92ζ74-ζ92ζ72-ζ9ζ74+ζ9ζ72 | -ζ98ζ75+ζ98ζ73+ζ97ζ76-ζ97ζ73+ζ94ζ75-ζ94ζ73+ζ92ζ76-ζ92ζ75 | -ζ98ζ76+ζ98ζ75-ζ97ζ75+ζ97ζ73+ζ94ζ76-ζ94ζ75-ζ92ζ76+ζ92ζ73 | ζ97ζ75-ζ97ζ73-ζ95ζ76+ζ95ζ73-ζ92ζ76+ζ92ζ75+ζ9ζ76-ζ9ζ73 | -ζ97ζ74+ζ97ζ72+ζ95ζ74-ζ95ζ7+ζ92ζ72-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 33 17 9 5 3)(4 34 49 25 13 7)(6 35 18 41 21 11)(8 36 50 57 29 15)(10 37 19)(12 38 51 26 45 23)(14 39 20 42 53 27)(16 40 52 58 61 31)(22 43)(24 44 54 59 30 47)(28 46 55)(32 48 56 60 62 63)
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), (2,33,17,9,5,3)(4,34,49,25,13,7)(6,35,18,41,21,11)(8,36,50,57,29,15)(10,37,19)(12,38,51,26,45,23)(14,39,20,42,53,27)(16,40,52,58,61,31)(22,43)(24,44,54,59,30,47)(28,46,55)(32,48,56,60,62,63)>;
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), (2,33,17,9,5,3)(4,34,49,25,13,7)(6,35,18,41,21,11)(8,36,50,57,29,15)(10,37,19)(12,38,51,26,45,23)(14,39,20,42,53,27)(16,40,52,58,61,31)(22,43)(24,44,54,59,30,47)(28,46,55)(32,48,56,60,62,63) );
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)], [(2,33,17,9,5,3),(4,34,49,25,13,7),(6,35,18,41,21,11),(8,36,50,57,29,15),(10,37,19),(12,38,51,26,45,23),(14,39,20,42,53,27),(16,40,52,58,61,31),(22,43),(24,44,54,59,30,47),(28,46,55),(32,48,56,60,62,63)]])
Matrix representation of C63⋊C6 ►in GL6(𝔽2)
| 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(2))| [0,1,0,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,1,1,1,1],[1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,1,1,0,1,1,0,0,0,0,1,0] >;
C63⋊C6 in GAP, Magma, Sage, TeX
C_{63}\rtimes C_6 % in TeX
G:=Group("C63:C6"); // GroupNames label
G:=SmallGroup(378,13);
// by ID
G=gap.SmallGroup(378,13);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,2072,997,642,368,6304]);
// Polycyclic
G:=Group<a,b|a^63=b^6=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C63⋊C6 in TeX
Character table of C63⋊C6 in TeX