metabelian, soluble, monomial, A-group
Aliases: C72⋊C8, C7⋊D7.C4, C72⋊C4.1C2, SmallGroup(392,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72⋊C8 |
Generators and relations for C72⋊C8
G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a-1b2, cbc-1=ab4 >
Character table of C72⋊C8
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 7D | 7E | 7F | 8A | 8B | 8C | 8D | |
| size | 1 | 49 | 49 | 49 | 8 | 8 | 8 | 8 | 8 | 8 | 49 | 49 | 49 | 49 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ5 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ8 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ9 | 8 | 0 | 0 | 0 | -2ζ76-2ζ7-2 | ζ75+2ζ74+2ζ73+ζ72+2 | 2ζ76+ζ74+ζ73+2ζ7+2 | -2ζ74-2ζ73-2 | -2ζ75-2ζ72-2 | ζ76+2ζ75+2ζ72+ζ7+2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ10 | 8 | 0 | 0 | 0 | 2ζ76+ζ74+ζ73+2ζ7+2 | -2ζ76-2ζ7-2 | -2ζ75-2ζ72-2 | ζ75+2ζ74+2ζ73+ζ72+2 | ζ76+2ζ75+2ζ72+ζ7+2 | -2ζ74-2ζ73-2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ11 | 8 | 0 | 0 | 0 | ζ75+2ζ74+2ζ73+ζ72+2 | -2ζ74-2ζ73-2 | -2ζ76-2ζ7-2 | ζ76+2ζ75+2ζ72+ζ7+2 | 2ζ76+ζ74+ζ73+2ζ7+2 | -2ζ75-2ζ72-2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ12 | 8 | 0 | 0 | 0 | ζ76+2ζ75+2ζ72+ζ7+2 | -2ζ75-2ζ72-2 | -2ζ74-2ζ73-2 | 2ζ76+ζ74+ζ73+2ζ7+2 | ζ75+2ζ74+2ζ73+ζ72+2 | -2ζ76-2ζ7-2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ13 | 8 | 0 | 0 | 0 | -2ζ74-2ζ73-2 | ζ76+2ζ75+2ζ72+ζ7+2 | ζ75+2ζ74+2ζ73+ζ72+2 | -2ζ75-2ζ72-2 | -2ζ76-2ζ7-2 | 2ζ76+ζ74+ζ73+2ζ7+2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 8 | 0 | 0 | 0 | -2ζ75-2ζ72-2 | 2ζ76+ζ74+ζ73+2ζ7+2 | ζ76+2ζ75+2ζ72+ζ7+2 | -2ζ76-2ζ7-2 | -2ζ74-2ζ73-2 | ζ75+2ζ74+2ζ73+ζ72+2 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 28 points - transitive group 28T56
Generators in S28
(1 17 24 8 12 28 13)(2 9 14 25 21 18 5)(3 19 26 10 6 22 15)
(1 24 12 13 17 8 28)(2 25 5 14 18 9 21)(3 6 19 22 26 15 10)(4 11 16 27 23 20 7)
(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)| (1,17,24,8,12,28,13)(2,9,14,25,21,18,5)(3,19,26,10,6,22,15), (1,24,12,13,17,8,28)(2,25,5,14,18,9,21)(3,6,19,22,26,15,10)(4,11,16,27,23,20,7), (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( (1,17,24,8,12,28,13)(2,9,14,25,21,18,5)(3,19,26,10,6,22,15), (1,24,12,13,17,8,28)(2,25,5,14,18,9,21)(3,6,19,22,26,15,10)(4,11,16,27,23,20,7), (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([[(1,17,24,8,12,28,13),(2,9,14,25,21,18,5),(3,19,26,10,6,22,15)], [(1,24,12,13,17,8,28),(2,25,5,14,18,9,21),(3,6,19,22,26,15,10),(4,11,16,27,23,20,7)], [(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(28,56);
Matrix representation of C72⋊C8 ►in GL8(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 24 | 112 | 112 | 24 | 0 | 0 | 0 | 0 |
| 73 | 77 | 0 | 0 | 10 | 89 | 0 | 0 |
| 21 | 98 | 0 | 0 | 24 | 112 | 0 | 0 |
| 40 | 36 | 0 | 0 | 0 | 0 | 10 | 89 |
| 92 | 15 | 0 | 0 | 0 | 0 | 24 | 112 |
| 112 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 87 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 88 | 89 | 10 | 0 | 0 | 0 | 0 |
| 50 | 78 | 103 | 103 | 0 | 0 | 0 | 0 |
| 36 | 98 | 0 | 0 | 24 | 112 | 0 | 0 |
| 15 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 25 | 36 | 0 | 0 | 0 | 0 | 89 | 10 |
| 41 | 73 | 0 | 0 | 0 | 0 | 103 | 103 |
| 0 | 0 | 0 | 0 | 112 | 1 | 0 | 0 |
| 21 | 98 | 0 | 0 | 111 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 0 |
| 88 | 1 | 0 | 0 | 98 | 0 | 0 | 0 |
G:=sub<GL(8,GF(113))| [1,0,0,24,73,21,40,92,0,1,0,112,77,98,36,15,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112],[112,87,36,50,36,15,25,41,1,25,88,78,98,0,36,73,0,0,89,103,0,0,0,0,0,0,10,103,0,0,0,0,0,0,0,0,24,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,89,103,0,0,0,0,0,0,10,103],[0,21,0,0,0,0,0,88,0,98,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,112,111,1,1,15,15,98,98,1,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C72⋊C8 in GAP, Magma, Sage, TeX
C_7^2\rtimes C_8
% in TeX
G:=Group("C7^2:C8"); // GroupNames label
G:=SmallGroup(392,36);
// by ID
G=gap.SmallGroup(392,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,7,10,26,6243,888,253,9604,2509,2114]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a*b^4>;
// generators/relations
Export
Subgroup lattice of C72⋊C8 in TeX
Character table of C72⋊C8 in TeX