direct product, cyclic, abelian, monomial
Aliases: C28, also denoted Z28, SmallGroup(28,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C28 |
| C1 — C28 |
| C1 — C28 |
Generators and relations for C28
G = < a | a28=1 >
Character table of C28
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 7D | 7E | 7F | 14A | 14B | 14C | 14D | 14E | 14F | 28A | 28B | 28C | 28D | 28E | 28F | 28G | 28H | 28I | 28J | 28K | 28L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -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 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | i | -i | i | -i | i | -i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | -i | i | -i | i | -i | i | -i | i | i | linear of order 4 |
| ρ5 | 1 | 1 | 1 | 1 | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | ζ76 | ζ74 | ζ7 | ζ76 | ζ7 | ζ72 | ζ72 | ζ73 | ζ73 | ζ74 | ζ74 | ζ75 | ζ75 | ζ76 | ζ7 | linear of order 7 |
| ρ6 | 1 | -1 | -i | i | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | -ζ73 | -ζ75 | -ζ72 | -ζ76 | -ζ74 | -ζ7 | ζ4ζ76 | ζ4ζ7 | ζ43ζ72 | ζ4ζ72 | ζ43ζ73 | ζ4ζ73 | ζ43ζ74 | ζ4ζ74 | ζ43ζ75 | ζ4ζ75 | ζ43ζ76 | ζ43ζ7 | linear of order 28 faithful |
| ρ7 | 1 | 1 | -1 | -1 | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | ζ76 | ζ74 | ζ7 | -ζ76 | -ζ7 | -ζ72 | -ζ72 | -ζ73 | -ζ73 | -ζ74 | -ζ74 | -ζ75 | -ζ75 | -ζ76 | -ζ7 | linear of order 14 |
| ρ8 | 1 | -1 | i | -i | ζ72 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | -ζ73 | -ζ75 | -ζ72 | -ζ76 | -ζ74 | -ζ7 | ζ43ζ76 | ζ43ζ7 | ζ4ζ72 | ζ43ζ72 | ζ4ζ73 | ζ43ζ73 | ζ4ζ74 | ζ43ζ74 | ζ4ζ75 | ζ43ζ75 | ζ4ζ76 | ζ4ζ7 | linear of order 28 faithful |
| ρ9 | 1 | 1 | 1 | 1 | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | ζ75 | ζ7 | ζ72 | ζ75 | ζ72 | ζ74 | ζ74 | ζ76 | ζ76 | ζ7 | ζ7 | ζ73 | ζ73 | ζ75 | ζ72 | linear of order 7 |
| ρ10 | 1 | -1 | -i | i | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | -ζ76 | -ζ73 | -ζ74 | -ζ75 | -ζ7 | -ζ72 | ζ4ζ75 | ζ4ζ72 | ζ43ζ74 | ζ4ζ74 | ζ43ζ76 | ζ4ζ76 | ζ43ζ7 | ζ4ζ7 | ζ43ζ73 | ζ4ζ73 | ζ43ζ75 | ζ43ζ72 | linear of order 28 faithful |
| ρ11 | 1 | 1 | -1 | -1 | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | ζ75 | ζ7 | ζ72 | -ζ75 | -ζ72 | -ζ74 | -ζ74 | -ζ76 | -ζ76 | -ζ7 | -ζ7 | -ζ73 | -ζ73 | -ζ75 | -ζ72 | linear of order 14 |
| ρ12 | 1 | -1 | i | -i | ζ74 | ζ76 | ζ7 | ζ73 | ζ75 | ζ72 | -ζ76 | -ζ73 | -ζ74 | -ζ75 | -ζ7 | -ζ72 | ζ43ζ75 | ζ43ζ72 | ζ4ζ74 | ζ43ζ74 | ζ4ζ76 | ζ43ζ76 | ζ4ζ7 | ζ43ζ7 | ζ4ζ73 | ζ43ζ73 | ζ4ζ75 | ζ4ζ72 | linear of order 28 faithful |
| ρ13 | 1 | 1 | 1 | 1 | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | ζ74 | ζ75 | ζ73 | ζ74 | ζ73 | ζ76 | ζ76 | ζ72 | ζ72 | ζ75 | ζ75 | ζ7 | ζ7 | ζ74 | ζ73 | linear of order 7 |
| ρ14 | 1 | -1 | -i | i | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | -ζ72 | -ζ7 | -ζ76 | -ζ74 | -ζ75 | -ζ73 | ζ4ζ74 | ζ4ζ73 | ζ43ζ76 | ζ4ζ76 | ζ43ζ72 | ζ4ζ72 | ζ43ζ75 | ζ4ζ75 | ζ43ζ7 | ζ4ζ7 | ζ43ζ74 | ζ43ζ73 | linear of order 28 faithful |
| ρ15 | 1 | 1 | -1 | -1 | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | ζ74 | ζ75 | ζ73 | -ζ74 | -ζ73 | -ζ76 | -ζ76 | -ζ72 | -ζ72 | -ζ75 | -ζ75 | -ζ7 | -ζ7 | -ζ74 | -ζ73 | linear of order 14 |
| ρ16 | 1 | -1 | i | -i | ζ76 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | -ζ72 | -ζ7 | -ζ76 | -ζ74 | -ζ75 | -ζ73 | ζ43ζ74 | ζ43ζ73 | ζ4ζ76 | ζ43ζ76 | ζ4ζ72 | ζ43ζ72 | ζ4ζ75 | ζ43ζ75 | ζ4ζ7 | ζ43ζ7 | ζ4ζ74 | ζ4ζ73 | linear of order 28 faithful |
| ρ17 | 1 | 1 | 1 | 1 | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | ζ73 | ζ72 | ζ74 | ζ73 | ζ74 | ζ7 | ζ7 | ζ75 | ζ75 | ζ72 | ζ72 | ζ76 | ζ76 | ζ73 | ζ74 | linear of order 7 |
| ρ18 | 1 | -1 | -i | i | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | -ζ75 | -ζ76 | -ζ7 | -ζ73 | -ζ72 | -ζ74 | ζ4ζ73 | ζ4ζ74 | ζ43ζ7 | ζ4ζ7 | ζ43ζ75 | ζ4ζ75 | ζ43ζ72 | ζ4ζ72 | ζ43ζ76 | ζ4ζ76 | ζ43ζ73 | ζ43ζ74 | linear of order 28 faithful |
| ρ19 | 1 | 1 | -1 | -1 | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | ζ75 | ζ76 | ζ7 | ζ73 | ζ72 | ζ74 | -ζ73 | -ζ74 | -ζ7 | -ζ7 | -ζ75 | -ζ75 | -ζ72 | -ζ72 | -ζ76 | -ζ76 | -ζ73 | -ζ74 | linear of order 14 |
| ρ20 | 1 | -1 | i | -i | ζ7 | ζ75 | ζ72 | ζ76 | ζ73 | ζ74 | -ζ75 | -ζ76 | -ζ7 | -ζ73 | -ζ72 | -ζ74 | ζ43ζ73 | ζ43ζ74 | ζ4ζ7 | ζ43ζ7 | ζ4ζ75 | ζ43ζ75 | ζ4ζ72 | ζ43ζ72 | ζ4ζ76 | ζ43ζ76 | ζ4ζ73 | ζ4ζ74 | linear of order 28 faithful |
| ρ21 | 1 | 1 | 1 | 1 | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | ζ72 | ζ76 | ζ75 | ζ72 | ζ75 | ζ73 | ζ73 | ζ7 | ζ7 | ζ76 | ζ76 | ζ74 | ζ74 | ζ72 | ζ75 | linear of order 7 |
| ρ22 | 1 | -1 | -i | i | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | -ζ7 | -ζ74 | -ζ73 | -ζ72 | -ζ76 | -ζ75 | ζ4ζ72 | ζ4ζ75 | ζ43ζ73 | ζ4ζ73 | ζ43ζ7 | ζ4ζ7 | ζ43ζ76 | ζ4ζ76 | ζ43ζ74 | ζ4ζ74 | ζ43ζ72 | ζ43ζ75 | linear of order 28 faithful |
| ρ23 | 1 | 1 | -1 | -1 | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | ζ7 | ζ74 | ζ73 | ζ72 | ζ76 | ζ75 | -ζ72 | -ζ75 | -ζ73 | -ζ73 | -ζ7 | -ζ7 | -ζ76 | -ζ76 | -ζ74 | -ζ74 | -ζ72 | -ζ75 | linear of order 14 |
| ρ24 | 1 | -1 | i | -i | ζ73 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | -ζ7 | -ζ74 | -ζ73 | -ζ72 | -ζ76 | -ζ75 | ζ43ζ72 | ζ43ζ75 | ζ4ζ73 | ζ43ζ73 | ζ4ζ7 | ζ43ζ7 | ζ4ζ76 | ζ43ζ76 | ζ4ζ74 | ζ43ζ74 | ζ4ζ72 | ζ4ζ75 | linear of order 28 faithful |
| ρ25 | 1 | 1 | 1 | 1 | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | ζ7 | ζ73 | ζ76 | ζ7 | ζ76 | ζ75 | ζ75 | ζ74 | ζ74 | ζ73 | ζ73 | ζ72 | ζ72 | ζ7 | ζ76 | linear of order 7 |
| ρ26 | 1 | -1 | -i | i | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | -ζ74 | -ζ72 | -ζ75 | -ζ7 | -ζ73 | -ζ76 | ζ4ζ7 | ζ4ζ76 | ζ43ζ75 | ζ4ζ75 | ζ43ζ74 | ζ4ζ74 | ζ43ζ73 | ζ4ζ73 | ζ43ζ72 | ζ4ζ72 | ζ43ζ7 | ζ43ζ76 | linear of order 28 faithful |
| ρ27 | 1 | 1 | -1 | -1 | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | ζ74 | ζ72 | ζ75 | ζ7 | ζ73 | ζ76 | -ζ7 | -ζ76 | -ζ75 | -ζ75 | -ζ74 | -ζ74 | -ζ73 | -ζ73 | -ζ72 | -ζ72 | -ζ7 | -ζ76 | linear of order 14 |
| ρ28 | 1 | -1 | i | -i | ζ75 | ζ74 | ζ73 | ζ72 | ζ7 | ζ76 | -ζ74 | -ζ72 | -ζ75 | -ζ7 | -ζ73 | -ζ76 | ζ43ζ7 | ζ43ζ76 | ζ4ζ75 | ζ43ζ75 | ζ4ζ74 | ζ43ζ74 | ζ4ζ73 | ζ43ζ73 | ζ4ζ72 | ζ43ζ72 | ζ4ζ7 | ζ4ζ76 | linear of order 28 faithful |
►Regular action on 28 points - transitive group 28T1
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)
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)>;
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) );
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)]])
G:=TransitiveGroup(28,1);
C28 is a maximal subgroup of
C7⋊C8 Dic14 D28
| action | f(x) | Disc(f) |
|---|---|---|
| 28T1 | x28+x27+x26+x25+x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | 2927 |
Matrix representation of C28 ►in GL1(𝔽29) generated by
| 26 |
G:=sub<GL(1,GF(29))| [26] >;
C28 in GAP, Magma, Sage, TeX
C_{28} % in TeX
G:=Group("C28"); // GroupNames label
G:=SmallGroup(28,2);
// by ID
G=gap.SmallGroup(28,2);
# by ID
G:=PCGroup([3,-2,-7,-2,42]);
// Polycyclic
G:=Group<a|a^28=1>;
// generators/relations
Export
Subgroup lattice of C28 in TeX
Character table of C28 in TeX