p-group, cyclic, abelian, monomial
Aliases: C16, also denoted Z16, SmallGroup(16,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C16 |
| C1 — C16 |
| C1 — C16 |
Generators and relations for C16
G = < a | a16=1 >
Character table of C16
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 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 | trivial |
| ρ2 | 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 | ζ162 | ζ1610 | ζ166 | ζ1614 | ζ1615 | ζ169 | ζ165 | ζ1611 | ζ163 | ζ1613 | ζ167 | ζ16 | linear of order 16 faithful |
| ρ4 | 1 | 1 | -1 | -1 | i | i | -i | -i | ζ87 | ζ8 | ζ85 | ζ83 | ζ83 | ζ85 | ζ87 | ζ8 | linear of order 8 |
| ρ5 | 1 | -1 | i | -i | ζ166 | ζ1614 | ζ162 | ζ1610 | ζ1613 | ζ1611 | ζ1615 | ζ16 | ζ169 | ζ167 | ζ165 | ζ163 | linear of order 16 faithful |
| ρ6 | 1 | -1 | -i | i | ζ1610 | ζ162 | ζ1614 | ζ166 | ζ1611 | ζ1613 | ζ169 | ζ167 | ζ1615 | ζ16 | ζ163 | ζ165 | linear of order 16 faithful |
| ρ7 | 1 | 1 | -1 | -1 | -i | -i | i | i | ζ85 | ζ83 | ζ87 | ζ8 | ζ8 | ζ87 | ζ85 | ζ83 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | ζ1614 | ζ166 | ζ1610 | ζ162 | ζ169 | ζ1615 | ζ163 | ζ1613 | ζ165 | ζ1611 | ζ16 | ζ167 | linear of order 16 faithful |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ10 | 1 | -1 | -i | i | ζ162 | ζ1610 | ζ166 | ζ1614 | ζ167 | ζ16 | ζ1613 | ζ163 | ζ1611 | ζ165 | ζ1615 | ζ169 | linear of order 16 faithful |
| ρ11 | 1 | 1 | -1 | -1 | i | i | -i | -i | ζ83 | ζ85 | ζ8 | ζ87 | ζ87 | ζ8 | ζ83 | ζ85 | linear of order 8 |
| ρ12 | 1 | -1 | i | -i | ζ166 | ζ1614 | ζ162 | ζ1610 | ζ165 | ζ163 | ζ167 | ζ169 | ζ16 | ζ1615 | ζ1613 | ζ1611 | linear of order 16 faithful |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ14 | 1 | -1 | -i | i | ζ1610 | ζ162 | ζ1614 | ζ166 | ζ163 | ζ165 | ζ16 | ζ1615 | ζ167 | ζ169 | ζ1611 | ζ1613 | linear of order 16 faithful |
| ρ15 | 1 | 1 | -1 | -1 | -i | -i | i | i | ζ8 | ζ87 | ζ83 | ζ85 | ζ85 | ζ83 | ζ8 | ζ87 | linear of order 8 |
| ρ16 | 1 | -1 | i | -i | ζ1614 | ζ166 | ζ1610 | ζ162 | ζ16 | ζ167 | ζ1611 | ζ165 | ζ1613 | ζ163 | ζ169 | ζ1615 | linear of order 16 faithful |
►Regular action on 16 points - transitive group 16T1
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,1);
C16 is a maximal subgroup of
C32 D16 SD32 Q32 C32⋊2C16 C2.F9 F17 C52⋊C16
C2p.C8: M5(2) C3⋊C16 C5⋊2C16 C5⋊C16 C7⋊C16 C11⋊C16 C13⋊2C16 C13⋊C16 ...
C16 is a maximal quotient of
C32 C32⋊2C16 C2.F9 C52⋊C16
Cp⋊C16: C3⋊C16 C5⋊2C16 C5⋊C16 C7⋊C16 C11⋊C16 C13⋊2C16 C13⋊C16 C17⋊4C16 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 16T1 | x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | 1715 |
Matrix representation of C16 ►in GL1(𝔽17) generated by
| 7 |
G:=sub<GL(1,GF(17))| [7] >;
C16 in GAP, Magma, Sage, TeX
C_{16} % in TeX
G:=Group("C16"); // GroupNames label
G:=SmallGroup(16,1);
// by ID
G=gap.SmallGroup(16,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-2,8,21,34]);
// Polycyclic
G:=Group<a|a^16=1>;
// generators/relations
Export
Subgroup lattice of C16 in TeX
Character table of C16 in TeX