Finite Group Databasebeta

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C₂×C₃²×C₄×C₇×C₃₁ #72193645

order	15624
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 69120
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂×C₃

Aliases

abelian	C₂×C₃²×C₄×C₇×C₃₁
multiplicative	(ℤ/16039)ˣ, (ℤ/16939)ˣ, (ℤ/18277)ˣ, (ℤ/32078)ˣ, (ℤ/33878)ˣ, (ℤ/36554)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃²×C₇×C₃₁
socle, prime radical	C₂²×C₃²×C₇×C₃₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

192 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

15624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃²×C₇×C₈×C₃₁, C₃²×C₄²×C₇×C₃₁, C₂²×C₃²×C₄×C₇×C₃₁
maximal normal subgroup of	C₂×C₃²×C₇×C₈×C₃₁, C₃²×C₄²×C₇×C₃₁, C₂²×C₃²×C₄×C₇×C₃₁
maximal quotient of	C₂×C₃²×C₇×C₈×C₃₁, C₃²×C₄²×C₇×C₃₁, C₂²×C₃²×C₄×C₇×C₃₁
automorphism group of	C₄₃×C₃₇₃, C₁₃×C₁₃₀₃
outer automorphism group of	C₄₃×C₃₇₃, C₁₃×C₁₃₀₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.