metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic7, C7⋊C4, C2.D7, C14.C2, SmallGroup(28,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — Dic7 |
Generators and relations for Dic7
G = < a,b | a14=1, b2=a7, bab-1=a-1 >
Character table of Dic7
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ7 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ8 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic faithful, Schur index 2 |
| ρ9 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic faithful, Schur index 2 |
| ρ10 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic faithful, Schur index 2 |
►Regular action on 28 points - transitive group 28T3
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)
(1 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)
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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23)>;
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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23) );
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)], [(1,22,8,15),(2,21,9,28),(3,20,10,27),(4,19,11,26),(5,18,12,25),(6,17,13,24),(7,16,14,23)]])
G:=TransitiveGroup(28,3);
Dic7 is a maximal subgroup of
C4×D7 C7⋊D4 C7⋊C12 C7⋊F5 C7⋊Dic7 C32⋊Dic7 C91⋊C4 C17⋊Dic7
Dic7p: Dic14 Dic21 Dic35 Dic49 Dic77 Dic91 Dic119 ...
Dic7 is a maximal quotient of
C7⋊F5 C32⋊Dic7 C91⋊C4 C17⋊Dic7
C2p.D7: C7⋊C8 Dic21 Dic35 Dic49 C7⋊Dic7 Dic77 Dic91 Dic119 ...
Matrix representation of Dic7 ►in GL2(𝔽13) generated by
| 5 | 11 |
| 11 | 1 |
| 8 | 3 |
| 0 | 5 |
G:=sub<GL(2,GF(13))| [5,11,11,1],[8,0,3,5] >;
Dic7 in GAP, Magma, Sage, TeX
{\rm Dic}_7 % in TeX
G:=Group("Dic7"); // GroupNames label
G:=SmallGroup(28,1);
// by ID
G=gap.SmallGroup(28,1);
# by ID
G:=PCGroup([3,-2,-2,-7,6,218]);
// Polycyclic
G:=Group<a,b|a^14=1,b^2=a^7,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic7 in TeX
Character table of Dic7 in TeX