non-abelian, soluble, monomial
Aliases: C72⋊Q8, C72⋊C4.2C2, C7⋊D7.2C22, SmallGroup(392,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊Q8
G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, cac-1=a4b2, dad-1=a3b-1, cbc-1=a2b3, dbd-1=a3b4, dcd-1=c-1 >
Character table of C72⋊Q8
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 7D | 7E | 7F | |
| size | 1 | 49 | 98 | 98 | 98 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ6 | 8 | 0 | 0 | 0 | 0 | 3ζ75+ζ74+ζ73+3ζ72 | ζ76+3ζ74+3ζ73+ζ7 | 1 | 1 | 1 | 3ζ76+ζ75+ζ72+3ζ7 | orthogonal faithful |
| ρ7 | 8 | 0 | 0 | 0 | 0 | 3ζ76+ζ75+ζ72+3ζ7 | 3ζ75+ζ74+ζ73+3ζ72 | 1 | 1 | 1 | ζ76+3ζ74+3ζ73+ζ7 | orthogonal faithful |
| ρ8 | 8 | 0 | 0 | 0 | 0 | 1 | 1 | 3ζ76+ζ75+ζ72+3ζ7 | 3ζ75+ζ74+ζ73+3ζ72 | ζ76+3ζ74+3ζ73+ζ7 | 1 | orthogonal faithful |
| ρ9 | 8 | 0 | 0 | 0 | 0 | 1 | 1 | 3ζ75+ζ74+ζ73+3ζ72 | ζ76+3ζ74+3ζ73+ζ7 | 3ζ76+ζ75+ζ72+3ζ7 | 1 | orthogonal faithful |
| ρ10 | 8 | 0 | 0 | 0 | 0 | 1 | 1 | ζ76+3ζ74+3ζ73+ζ7 | 3ζ76+ζ75+ζ72+3ζ7 | 3ζ75+ζ74+ζ73+3ζ72 | 1 | orthogonal faithful |
| ρ11 | 8 | 0 | 0 | 0 | 0 | ζ76+3ζ74+3ζ73+ζ7 | 3ζ76+ζ75+ζ72+3ζ7 | 1 | 1 | 1 | 3ζ75+ζ74+ζ73+3ζ72 | orthogonal faithful |
►On 28 points - transitive group 28T58
Generators in S28
(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)
(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)(22 24 26 28 23 25 27)
(1 9)(2 11 7 14)(3 13 6 12)(4 8 5 10)(15 27 16 26)(17 25 21 28)(18 24 20 22)(19 23)
(1 19)(2 17 7 21)(3 15 6 16)(4 20 5 18)(8 24 10 22)(9 23)(11 28 14 25)(12 27 13 26)
G:=sub<Sym(28)| (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), (8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27), (1,9)(2,11,7,14)(3,13,6,12)(4,8,5,10)(15,27,16,26)(17,25,21,28)(18,24,20,22)(19,23), (1,19)(2,17,7,21)(3,15,6,16)(4,20,5,18)(8,24,10,22)(9,23)(11,28,14,25)(12,27,13,26)>;
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), (8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27), (1,9)(2,11,7,14)(3,13,6,12)(4,8,5,10)(15,27,16,26)(17,25,21,28)(18,24,20,22)(19,23), (1,19)(2,17,7,21)(3,15,6,16)(4,20,5,18)(8,24,10,22)(9,23)(11,28,14,25)(12,27,13,26) );
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)], [(8,12,9,13,10,14,11),(15,16,17,18,19,20,21),(22,24,26,28,23,25,27)], [(1,9),(2,11,7,14),(3,13,6,12),(4,8,5,10),(15,27,16,26),(17,25,21,28),(18,24,20,22),(19,23)], [(1,19),(2,17,7,21),(3,15,6,16),(4,20,5,18),(8,24,10,22),(9,23),(11,28,14,25),(12,27,13,26)]])
G:=TransitiveGroup(28,58);
Matrix representation of C72⋊Q8 ►in GL8(𝔽29)
| 28 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 10 | 11 | 4 | 0 | 0 | 0 | 0 |
| 9 | 6 | 25 | 25 | 0 | 0 | 0 | 0 |
| 20 | 23 | 0 | 0 | 25 | 25 | 0 | 0 |
| 5 | 19 | 0 | 0 | 4 | 11 | 0 | 0 |
| 24 | 10 | 0 | 0 | 0 | 0 | 11 | 4 |
| 9 | 6 | 0 | 0 | 0 | 0 | 25 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 10 | 4 | 11 | 0 | 0 | 0 | 0 |
| 18 | 28 | 18 | 28 | 0 | 0 | 0 | 0 |
| 21 | 23 | 0 | 0 | 25 | 25 | 0 | 0 |
| 6 | 19 | 0 | 0 | 4 | 11 | 0 | 0 |
| 18 | 28 | 0 | 0 | 0 | 0 | 18 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 | 0 | 0 |
| 18 | 28 | 27 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 1 | 0 |
| 18 | 28 | 28 | 0 | 0 | 0 | 18 | 28 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 | 0 | 0 |
| 11 | 1 | 0 | 0 | 27 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 18 | 28 | 18 | 28 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(29))| [28,9,24,9,20,5,24,9,1,19,10,6,23,19,10,6,0,0,11,25,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,11,25,0,0,0,0,0,0,4,25],[1,0,23,18,21,6,18,0,0,1,10,28,23,19,28,0,0,0,4,18,0,0,0,0,0,0,11,28,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,28,0],[0,18,0,10,0,18,0,0,0,28,0,1,0,28,0,0,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,18,0,0,0,0,0,0,0,28,0,0],[0,11,0,0,0,10,0,18,0,1,0,0,0,1,0,28,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,28,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C72⋊Q8 in GAP, Magma, Sage, TeX
C_7^2\rtimes Q_8
% in TeX
G:=Group("C7^2:Q8"); // GroupNames label
G:=SmallGroup(392,38);
// by ID
G=gap.SmallGroup(392,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,7,20,61,26,1763,3048,253,5004,3309,2114]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^7=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^4*b^2,d*a*d^-1=a^3*b^-1,c*b*c^-1=a^2*b^3,d*b*d^-1=a^3*b^4,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C72⋊Q8 in TeX
Character table of C72⋊Q8 in TeX