Finite Group Databasebeta

← Prev (#28) Next (#30) →

C₁₆ #29

order	16
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄ (order 8) · Inn(G) = C₁ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/16)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₆
multiplicative	(ℤ/17)ˣ, (ℤ/34)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂
socle, prime radical	C₂	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

5 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 1 generator on 16 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

center of	M₆₄, #504, C₃⋊C₃₂, …
derived subgroup of	D₃₂, SD₆₄, Q₆₄, …
Frattini subgroup of	C₃₂, C₂×C₃₂, M₆₄, …
Fitting subgroup of	C₁₆×A₅, #11746446, A₅⋊C₃₂, …
maximal subgroup of	C₃₂, C₂×C₁₆, M₃₂, …
maximal normal subgroup of	C₃₂, C₂×C₁₆, M₃₂, …
maximal quotient of	C₃₂, C₂×C₁₆, C₃⋊C₁₆, …
automorphism group of	C₁₇, C₂×C₁₇
outer automorphism group of	C₁₇, C₂×C₁₇, D₃×C₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 1);