C16:C4 - GroupNames

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G = C₁₆⋊C₄ order 64 = 2⁶

2^nd semidirect product of C₁₆ and C₄ acting faithfully

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C₁₆⋊₂C₄, C₄².₁C₄, C₄.₁₁C₄², C₄.₆M₄(2), M₅(2).₂C₂, C₂².₄M₄(2), (C₂×C₈).₂C₄, C₈.₂₀(C₂×C₄), C₈⋊C₄.₄C₂, C₂.₃(C₈⋊C₄), (C₂×C₈).₄₁C₂², (C₂×C₄).₆₆(C₂×C₄), SmallGroup(64,28)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

C₁ — C₄ — C₁₆⋊C₄

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₈⋊C₄ — C₁₆⋊C₄

C₁ — C₄ — C₁₆⋊C₄

C₁ — C₄ — C₁₆⋊C₄

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₁₆⋊C₄

Generators and relations for C₁₆⋊C₄
G = < a,b | a¹⁶=b⁴=1, bab^-1=a¹³ >

C₁

C₂

₂C₂

C₄

C₂²

C₄

₄C₄

C₈

C₈

₂C₂×C₄

₂C₈

C₁₆

C₁₆

C₁₆

C₁₆

C₄²

Character table of C₁₆⋊C₄

class	1	2A	2B	4A	4B	4C	4D	4E	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	2	1	1	2	4	4	2	2	2	2	4	4	4	4	4	4	4	4	4	4
ρ₁	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	linear of order 2
ρ₃	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
ρ₄	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
ρ₅	1	1	-1	1	1	-1	-i	i	-1	-1	1	1	i	-i	i	-1	i	1	1	-i	-1	-i	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₇	1	1	-1	1	1	-1	i	-i	-1	-1	1	1	-i	i	-i	-1	-i	1	1	i	-1	i	linear of order 4
ρ₈	1	1	-1	1	1	-1	-i	i	-1	-1	1	1	i	-i	-i	1	-i	-1	-1	i	1	i	linear of order 4
ρ₉	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	-i	i	i	-i	i	i	-i	-i	linear of order 4
ρ₁₀	1	1	-1	1	1	-1	i	-i	-1	-1	1	1	-i	i	i	1	i	-1	-1	-i	1	-i	linear of order 4
ρ₁₁	1	1	-1	1	1	-1	i	-i	1	1	-1	-1	i	-i	1	-i	-1	-i	i	1	i	-1	linear of order 4
ρ₁₂	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	-i	i	i	-i	linear of order 4
ρ₁₃	1	1	-1	1	1	-1	-i	i	1	1	-1	-1	-i	i	-1	-i	1	-i	i	-1	i	1	linear of order 4
ρ₁₄	1	1	-1	1	1	-1	i	-i	1	1	-1	-1	i	-i	-1	i	1	i	-i	-1	-i	1	linear of order 4
ρ₁₅	1	1	-1	1	1	-1	-i	i	1	1	-1	-1	-i	i	1	i	-1	i	-i	1	-i	-1	linear of order 4
ρ₁₆	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	i	-i	-i	i	linear of order 4
ρ₁₇	2	2	-2	-2	-2	2	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₁₈	2	2	-2	-2	-2	2	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₁₉	2	2	2	-2	-2	-2	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₀	2	2	2	-2	-2	-2	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₁	4	-4	0	4i	-4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₂	4	-4	0	-4i	4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₁₆⋊C₄
►On 16 points - transitive group 16T125

Generators in S₁₆

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(2 6 10 14)(3 11)(4 16 12 8)(7 15)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,6,10,14)(3,11)(4,16,12,8)(7,15)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,6,10,14)(3,11)(4,16,12,8)(7,15) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15)]])

G:=TransitiveGroup(16,125);

C₁₆⋊C₄ is a maximal subgroup of
C₄.C₄≀C₂ C₄².(C₂×C₄) C₈.₅M₄(2) C₈.₁₉M₄(2) Q₃₂⋊C₄ D₁₆⋊C₄
C_4p.C₄²: C₈.₂₃C₄² C₁₂.₁₅C₄² C₄₈⋊C₄ C₂₀.₄₅C₄² C₈₀⋊C₄ C₁₆⋊F₅ C₁₆⋊₄F₅ C₄².₃F₅ ...
C₁₆⋊C₄ is a maximal quotient of
C₄².₂C₈ C₁₆⋊F₅ C₁₆⋊₄F₅
C_4p.M₄(2): C₁₆⋊C₈ C₁₂.₁₅C₄² C₄₈⋊C₄ C₂₀.₄₅C₄² C₈₀⋊C₄ C₄².₃F₅ C₂₀.₂₃C₄² C₂₈.₁₅C₄² ...

Matrix representation of C₁₆⋊C₄ ►in GL₄(𝔽₅) generated by

0	0	0	3
0	0	4	0
1	0	0	0
0	1	0	0

,

2	0	0	0
0	3	0	0
0	0	4	0
0	0	0	1

G:=sub<GL(4,GF(5))| [0,0,1,0,0,0,0,1,0,4,0,0,3,0,0,0],[2,0,0,0,0,3,0,0,0,0,4,0,0,0,0,1] >;

C₁₆⋊C₄ in GAP, Magma, Sage, TeX

C_{16}\rtimes C_4

% in TeX

G:=Group("C16:C4");

// GroupNames label

G:=SmallGroup(64,28);

// by ID

G=gap.SmallGroup(64,28);

# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,24,217,55,332,86,963,88]);

// Polycyclic

G:=Group<a,b|a^16=b^4=1,b*a*b^-1=a^13>;

// generators/relations

Export

Subgroup lattice of C₁₆⋊C₄ in TeX
Character table of C₁₆⋊C₄ in TeX

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