metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7⋊C9, C21.C3, C3.(C7⋊C3), SmallGroup(63,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C9 |
Generators and relations for C7⋊C9
G = < a,b | a7=b9=1, bab-1=a4 >
Character table of C7⋊C9
| class | 1 | 3A | 3B | 7A | 7B | 9A | 9B | 9C | 9D | 9E | 9F | 21A | 21B | 21C | 21D | |
| size | 1 | 1 | 1 | 3 | 3 | 7 | 7 | 7 | 7 | 7 | 7 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | ζ3 | ζ32 | 1 | 1 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ5 | 1 | ζ3 | ζ32 | 1 | 1 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ6 | 1 | ζ32 | ζ3 | 1 | 1 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ7 | 1 | ζ3 | ζ32 | 1 | 1 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ8 | 1 | ζ32 | ζ3 | 1 | 1 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ9 | 1 | ζ32 | ζ3 | 1 | 1 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ10 | 3 | 3 | 3 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ11 | 3 | 3 | 3 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ12 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | complex faithful, Schur index 3 |
| ρ13 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | complex faithful, Schur index 3 |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | complex faithful, Schur index 3 |
| ρ15 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ74+ζ32ζ72+ζ32ζ7 | ζ32ζ76+ζ32ζ75+ζ32ζ73 | ζ3ζ76+ζ3ζ75+ζ3ζ73 | ζ3ζ74+ζ3ζ72+ζ3ζ7 | complex faithful, Schur index 3 |
►Regular action on 63 points
Generators in S63
(1 56 51 34 42 17 19)(2 43 57 18 52 20 35)(3 53 44 21 58 36 10)(4 59 54 28 45 11 22)(5 37 60 12 46 23 29)(6 47 38 24 61 30 13)(7 62 48 31 39 14 25)(8 40 63 15 49 26 32)(9 50 41 27 55 33 16)
(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)
G:=sub<Sym(63)| (1,56,51,34,42,17,19)(2,43,57,18,52,20,35)(3,53,44,21,58,36,10)(4,59,54,28,45,11,22)(5,37,60,12,46,23,29)(6,47,38,24,61,30,13)(7,62,48,31,39,14,25)(8,40,63,15,49,26,32)(9,50,41,27,55,33,16), (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)>;
G:=Group( (1,56,51,34,42,17,19)(2,43,57,18,52,20,35)(3,53,44,21,58,36,10)(4,59,54,28,45,11,22)(5,37,60,12,46,23,29)(6,47,38,24,61,30,13)(7,62,48,31,39,14,25)(8,40,63,15,49,26,32)(9,50,41,27,55,33,16), (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) );
G=PermutationGroup([[(1,56,51,34,42,17,19),(2,43,57,18,52,20,35),(3,53,44,21,58,36,10),(4,59,54,28,45,11,22),(5,37,60,12,46,23,29),(6,47,38,24,61,30,13),(7,62,48,31,39,14,25),(8,40,63,15,49,26,32),(9,50,41,27,55,33,16)], [(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)]])
C7⋊C9 is a maximal subgroup of
C7⋊C18 C9×C7⋊C3 C63⋊C3 C63⋊3C3 C21.C32 C21.A4 C49⋊C9 C72⋊C9 C72⋊3C9
C7⋊C9 is a maximal quotient of
C7⋊C27 C21.A4 C49⋊C9 C72⋊C9 C72⋊3C9
Matrix representation of C7⋊C9 ►in GL3(𝔽127) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 23 | 22 |
| 49 | 2 | 53 |
| 111 | 53 | 85 |
| 53 | 125 | 25 |
G:=sub<GL(3,GF(127))| [0,0,1,1,0,23,0,1,22],[49,111,53,2,53,125,53,85,25] >;
C7⋊C9 in GAP, Magma, Sage, TeX
C_7\rtimes C_9
% in TeX
G:=Group("C7:C9"); // GroupNames label
G:=SmallGroup(63,1);
// by ID
G=gap.SmallGroup(63,1);
# by ID
G:=PCGroup([3,-3,-3,-7,9,164]);
// Polycyclic
G:=Group<a,b|a^7=b^9=1,b*a*b^-1=a^4>;
// generators/relations
Export
Subgroup lattice of C7⋊C9 in TeX
Character table of C7⋊C9 in TeX