metabelian, supersoluble, monomial, A-group
Aliases: C72⋊3C3, C7⋊2(C7⋊C3), SmallGroup(147,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72⋊3C3 |
Generators and relations for C72⋊3C3
G = < a,b,c | a7=b7=c3=1, ab=ba, cac-1=a4, cbc-1=b2 >
Character table of C72⋊3C3
| class | 1 | 3A | 3B | 7A | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 7J | 7K | 7L | 7M | 7N | 7O | 7P | |
| size | 1 | 49 | 49 | 3 | 3 | 3 | 3 | 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 | trivial |
| ρ2 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | -1-√-7/2 | ζ76+2ζ74 | ζ73+2ζ72 | ζ75+ζ72+1 | ζ75+2ζ7 | ζ74+ζ73+1 | ζ76+ζ7+1 | -1+√-7/2 | ζ75+ζ72+1 | ζ76+ζ7+1 | 2ζ75+ζ74 | ζ74+ζ73+1 | 2ζ73+ζ7 | 2ζ76+ζ72 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ5 | 3 | 0 | 0 | -1-√-7/2 | ζ73+2ζ72 | ζ75+2ζ7 | ζ76+ζ7+1 | ζ76+2ζ74 | ζ75+ζ72+1 | ζ74+ζ73+1 | -1+√-7/2 | ζ76+ζ7+1 | ζ74+ζ73+1 | 2ζ76+ζ72 | ζ75+ζ72+1 | 2ζ75+ζ74 | 2ζ73+ζ7 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ6 | 3 | 0 | 0 | -1-√-7/2 | ζ74+ζ73+1 | ζ75+ζ72+1 | ζ76+2ζ74 | ζ76+ζ7+1 | ζ75+2ζ7 | ζ73+2ζ72 | -1+√-7/2 | 2ζ73+ζ7 | 2ζ75+ζ74 | ζ75+ζ72+1 | 2ζ76+ζ72 | ζ74+ζ73+1 | ζ76+ζ7+1 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ7 | 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 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | 3 | 3 | complex lifted from C7⋊C3 |
| ρ8 | 3 | 0 | 0 | -1+√-7/2 | ζ76+ζ7+1 | ζ74+ζ73+1 | 2ζ76+ζ72 | ζ75+ζ72+1 | 2ζ75+ζ74 | 2ζ73+ζ7 | -1-√-7/2 | ζ75+2ζ7 | ζ76+2ζ74 | ζ74+ζ73+1 | ζ73+2ζ72 | ζ76+ζ7+1 | ζ75+ζ72+1 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ9 | 3 | 0 | 0 | -1+√-7/2 | 2ζ75+ζ74 | 2ζ76+ζ72 | ζ76+ζ7+1 | 2ζ73+ζ7 | ζ75+ζ72+1 | ζ74+ζ73+1 | -1-√-7/2 | ζ76+ζ7+1 | ζ74+ζ73+1 | ζ75+2ζ7 | ζ75+ζ72+1 | ζ73+2ζ72 | ζ76+2ζ74 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ10 | 3 | 0 | 0 | -1+√-7/2 | 2ζ73+ζ7 | 2ζ75+ζ74 | ζ75+ζ72+1 | 2ζ76+ζ72 | ζ74+ζ73+1 | ζ76+ζ7+1 | -1-√-7/2 | ζ75+ζ72+1 | ζ76+ζ7+1 | ζ73+2ζ72 | ζ74+ζ73+1 | ζ76+2ζ74 | ζ75+2ζ7 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ11 | 3 | 0 | 0 | -1+√-7/2 | ζ74+ζ73+1 | ζ75+ζ72+1 | 2ζ73+ζ7 | ζ76+ζ7+1 | 2ζ76+ζ72 | 2ζ75+ζ74 | -1-√-7/2 | ζ76+2ζ74 | ζ73+2ζ72 | ζ75+ζ72+1 | ζ75+2ζ7 | ζ74+ζ73+1 | ζ76+ζ7+1 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ12 | 3 | 0 | 0 | -1-√-7/2 | ζ75+2ζ7 | ζ76+2ζ74 | ζ74+ζ73+1 | ζ73+2ζ72 | ζ76+ζ7+1 | ζ75+ζ72+1 | -1+√-7/2 | ζ74+ζ73+1 | ζ75+ζ72+1 | 2ζ73+ζ7 | ζ76+ζ7+1 | 2ζ76+ζ72 | 2ζ75+ζ74 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ13 | 3 | 0 | 0 | -1-√-7/2 | ζ75+ζ72+1 | ζ76+ζ7+1 | ζ73+2ζ72 | ζ74+ζ73+1 | ζ76+2ζ74 | ζ75+2ζ7 | -1+√-7/2 | 2ζ75+ζ74 | 2ζ76+ζ72 | ζ76+ζ7+1 | 2ζ73+ζ7 | ζ75+ζ72+1 | ζ74+ζ73+1 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ14 | 3 | 0 | 0 | 3 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | 3 | -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 |
| ρ15 | 3 | 0 | 0 | -1+√-7/2 | 2ζ76+ζ72 | 2ζ73+ζ7 | ζ74+ζ73+1 | 2ζ75+ζ74 | ζ76+ζ7+1 | ζ75+ζ72+1 | -1-√-7/2 | ζ74+ζ73+1 | ζ75+ζ72+1 | ζ76+2ζ74 | ζ76+ζ7+1 | ζ75+2ζ7 | ζ73+2ζ72 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ16 | 3 | 0 | 0 | -1-√-7/2 | ζ76+ζ7+1 | ζ74+ζ73+1 | ζ75+2ζ7 | ζ75+ζ72+1 | ζ73+2ζ72 | ζ76+2ζ74 | -1+√-7/2 | 2ζ76+ζ72 | 2ζ73+ζ7 | ζ74+ζ73+1 | 2ζ75+ζ74 | ζ76+ζ7+1 | ζ75+ζ72+1 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ17 | 3 | 0 | 0 | -1+√-7/2 | ζ75+ζ72+1 | ζ76+ζ7+1 | 2ζ75+ζ74 | ζ74+ζ73+1 | 2ζ73+ζ7 | 2ζ76+ζ72 | -1-√-7/2 | ζ73+2ζ72 | ζ75+2ζ7 | ζ76+ζ7+1 | ζ76+2ζ74 | ζ75+ζ72+1 | ζ74+ζ73+1 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ18 | 3 | 0 | 0 | 3 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | 3 | -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 |
| ρ19 | 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 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | 3 | 3 | complex lifted from C7⋊C3 |
►On 21 points - transitive group 21T12
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 3 5 7 2 4 6)(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)
(1 17 13)(2 19 10)(3 21 14)(4 16 11)(5 18 8)(6 20 12)(7 15 9)
G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,16,17,18,19,20,21), (1,17,13)(2,19,10)(3,21,14)(4,16,11)(5,18,8)(6,20,12)(7,15,9)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,16,17,18,19,20,21), (1,17,13)(2,19,10)(3,21,14)(4,16,11)(5,18,8)(6,20,12)(7,15,9) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,3,5,7,2,4,6),(8,12,9,13,10,14,11),(15,16,17,18,19,20,21)], [(1,17,13),(2,19,10),(3,21,14),(4,16,11),(5,18,8),(6,20,12),(7,15,9)]])
G:=TransitiveGroup(21,12);
C72⋊3C3 is a maximal subgroup of
C72⋊S3 C7⋊3F7 C72⋊C6 C7⋊C32
C72⋊3C3 is a maximal quotient of C72⋊3C9
Matrix representation of C72⋊3C3 ►in GL3(𝔽43) generated by
| 21 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 35 |
| 41 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(43))| [21,0,0,0,11,0,0,0,35],[41,0,0,0,16,0,0,0,4],[0,0,1,1,0,0,0,1,0] >;
C72⋊3C3 in GAP, Magma, Sage, TeX
C_7^2\rtimes_3C_3
% in TeX
G:=Group("C7^2:3C3"); // GroupNames label
G:=SmallGroup(147,5);
// by ID
G=gap.SmallGroup(147,5);
# by ID
G:=PCGroup([3,-3,-7,-7,37,758]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^3=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^2>;
// generators/relations
Export
Subgroup lattice of C72⋊3C3 in TeX
Character table of C72⋊3C3 in TeX