Finite Group Databasebeta

← Prev (#72182137) Next (#72182139) →

C₃×C₈×C₈₉ #72182138

order	2136
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₈×C₁₁ (order 704) · Inn(G) = C₁ · Out(G) = C₂³×C₈×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₁₃₆
abelian	C₃×C₈×C₈₉
multiplicative	(ℤ/2137)ˣ, (ℤ/4274)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃×C₈₉
socle, prime radical	C₂×C₃×C₈₉	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2136 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₁₆×C₈₉, C₂×C₃×C₈×C₈₉, C₈×C₉×C₈₉, …
maximal normal subgroup of	C₃×C₁₆×C₈₉, C₂×C₃×C₈×C₈₉, C₈×C₉×C₈₉, …
maximal quotient of	C₃×C₁₆×C₈₉, C₂×C₃×C₈×C₈₉, C₈×C₉×C₈₉, …
automorphism group of	C₂₁₃₇, C₂×C₂₁₃₇
outer automorphism group of	C₂₁₃₇, C₂×C₂₁₃₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.