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C₅×C₁₆×C₈₉ #72186167

order	7120
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄²×C₈×C₁₁ (order 2816) · Inn(G) = C₁ · Out(G) = C₂×C₄²×C₈×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₇₁₂₀
abelian	C₅×C₁₆×C₈₉
multiplicative	(ℤ/7121)ˣ, (ℤ/14242)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₅×C₈₉
socle, prime radical	C₂×C₅×C₈₉	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

7120 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	D₁₄₂₄₀, Dic₇₁₂₀
maximal subgroup of	C₅×C₃₂×C₈₉, C₂×C₅×C₁₆×C₈₉, C₃×C₅×C₁₆×C₈₉, …
maximal normal subgroup of	C₅×C₃₂×C₈₉, C₂×C₅×C₁₆×C₈₉, C₃×C₅×C₁₆×C₈₉, …
maximal quotient of	C₅×C₃₂×C₈₉, C₂×C₅×C₁₆×C₈₉, C₃×C₅×C₁₆×C₈₉
automorphism group of	C₇₁₂₁
outer automorphism group of	C₇₁₂₁

GAP

Past the SmallGroups library coverage — no SmallGroups id.