Finite Group Databasebeta

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C₅×C₁₆×C₁₆₇ #72191627

order	13360
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄²×C₈₃ (order 5312) · Inn(G) = C₁ · Out(G) = C₂²×C₄²×C₈₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₃₃₆₀
abelian	C₅×C₁₆×C₁₆₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₅×C₁₆₇
socle, prime radical	C₂×C₅×C₁₆₇	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

13360 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₃₃₆₀, D₂₆₇₂₀
Fitting subgroup of	D₁₃₃₆₀, Dic₆₆₈₀
maximal subgroup of	C₅×C₃₂×C₁₆₇, C₂×C₅×C₁₆×C₁₆₇, D₁₃₃₆₀, …
maximal normal subgroup of	C₅×C₃₂×C₁₆₇, C₂×C₅×C₁₆×C₁₆₇, D₁₃₃₆₀, …
maximal quotient of	C₅×C₃₂×C₁₆₇, C₂×C₅×C₁₆×C₁₆₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.