Finite Group Databasebeta

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C₂³×C₃×C₄×C₁₂₁ #72190057

order	11616
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 4730880
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁶

Aliases

abelian	C₂³×C₃×C₄×C₁₂₁
multiplicative	(ℤ/29080)ˣ, (ℤ/34872)ˣ, (ℤ/34896)ˣ, (ℤ/43620)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₁	C₂⁴×C₃×C₁₁
socle, prime radical	C₂⁴×C₃×C₁₁	C₂×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

708 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

11616 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×C₃×C₈×C₁₂₁, C₂²×C₃×C₄²×C₁₂₁, C₂⁴×C₃×C₄×C₁₂₁
maximal normal subgroup of	C₂³×C₃×C₈×C₁₂₁, C₂²×C₃×C₄²×C₁₂₁, C₂⁴×C₃×C₄×C₁₂₁
maximal quotient of	C₂³×C₃×C₈×C₁₂₁, C₂²×C₃×C₄²×C₁₂₁, C₂⁴×C₃×C₄×C₁₂₁
automorphism group of	C₅×C₈×C₇₂₇
outer automorphism group of	C₅×C₈×C₇₂₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.