metabelian, soluble, monomial, A-group
Aliases: C43⋊C7, C23.1F8, SmallGroup(448,178)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C7 |
Generators and relations for C43⋊C7
G = < a,b,c,d | a4=b4=c4=d7=1, ab=ba, ac=ca, dad-1=bc-1, bc=cb, dbd-1=a-1, dcd-1=b-1c2 >
Character table of C43⋊C7
| class | 1 | 2 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 7D | 7E | 7F | |
| size | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 64 | 64 | 64 | 64 | 64 | 64 | |
| ρ1 | 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 | ζ73 | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | linear of order 7 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ75 | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | linear of order 7 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ72 | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | linear of order 7 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ74 | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | linear of order 7 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ7 | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | linear of order 7 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ76 | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | linear of order 7 |
| ρ8 | 7 | 7 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F8 |
| ρ9 | 7 | -1 | 1-2i | 1+2i | 1+2i | -3-2i | -3+2i | 1-2i | 1+2i | 1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ10 | 7 | -1 | 1+2i | 1-2i | -3+2i | 1-2i | 1+2i | -3-2i | 1-2i | 1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 7 | -1 | 1-2i | 1+2i | -3-2i | 1+2i | 1-2i | -3+2i | 1+2i | 1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ12 | 7 | -1 | 1+2i | 1-2i | 1-2i | -3+2i | -3-2i | 1+2i | 1-2i | 1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 7 | -1 | -3-2i | -3+2i | 1-2i | 1-2i | 1+2i | 1+2i | 1-2i | 1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 7 | -1 | 1+2i | 1-2i | 1-2i | 1-2i | 1+2i | 1+2i | -3+2i | -3-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 7 | -1 | -3+2i | -3-2i | 1+2i | 1+2i | 1-2i | 1-2i | 1+2i | 1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 7 | -1 | 1-2i | 1+2i | 1+2i | 1+2i | 1-2i | 1-2i | -3-2i | -3+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 28 points - transitive group 28T61
Generators in S28
(1 20)(2 25 21 9)(3 15)(4 27 16 11)(5 28 17 12)(6 13 18 22)(7 19)(8 24)(10 26)(14 23)
(1 20)(2 21)(3 10 15 26)(4 16)(5 12 17 28)(6 13 18 22)(7 23 19 14)(8 24)(9 25)(11 27)
(1 24 20 8)(3 15)(4 27 16 11)(6 22 18 13)(7 14 19 23)(10 26)
(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)
G:=sub<Sym(28)| (1,20)(2,25,21,9)(3,15)(4,27,16,11)(5,28,17,12)(6,13,18,22)(7,19)(8,24)(10,26)(14,23), (1,20)(2,21)(3,10,15,26)(4,16)(5,12,17,28)(6,13,18,22)(7,23,19,14)(8,24)(9,25)(11,27), (1,24,20,8)(3,15)(4,27,16,11)(6,22,18,13)(7,14,19,23)(10,26), (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)>;
G:=Group( (1,20)(2,25,21,9)(3,15)(4,27,16,11)(5,28,17,12)(6,13,18,22)(7,19)(8,24)(10,26)(14,23), (1,20)(2,21)(3,10,15,26)(4,16)(5,12,17,28)(6,13,18,22)(7,23,19,14)(8,24)(9,25)(11,27), (1,24,20,8)(3,15)(4,27,16,11)(6,22,18,13)(7,14,19,23)(10,26), (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) );
G=PermutationGroup([[(1,20),(2,25,21,9),(3,15),(4,27,16,11),(5,28,17,12),(6,13,18,22),(7,19),(8,24),(10,26),(14,23)], [(1,20),(2,21),(3,10,15,26),(4,16),(5,12,17,28),(6,13,18,22),(7,23,19,14),(8,24),(9,25),(11,27)], [(1,24,20,8),(3,15),(4,27,16,11),(6,22,18,13),(7,14,19,23),(10,26)], [(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)]])
G:=TransitiveGroup(28,61);
Matrix representation of C43⋊C7 ►in GL7(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 23 | 0 | 17 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 12 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 26 | 0 | 0 | 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 12 | 0 | 0 | 0 | 0 | 0 |
| 23 | 0 | 17 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 28 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 1 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 1 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(7,GF(29))| [28,0,23,18,10,0,26,0,28,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,12],[28,27,23,16,0,1,0,0,12,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28],[12,26,0,16,0,17,7,0,17,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,27,22,9,5,6,13,4,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;
C43⋊C7 in GAP, Magma, Sage, TeX
C_4^3\rtimes C_7
% in TeX
G:=Group("C4^3:C7"); // GroupNames label
G:=SmallGroup(448,178);
// by ID
G=gap.SmallGroup(448,178);
# by ID
G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,792,590,352,7255,360,9804,16469,16470]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^7=1,a*b=b*a,a*c=c*a,d*a*d^-1=b*c^-1,b*c=c*b,d*b*d^-1=a^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
Export
Subgroup lattice of C43⋊C7 in TeX
Character table of C43⋊C7 in TeX