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C₂×C₃×C₄²×C₃₂ #72182811

order	3072
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 50331648
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³×C₄³

Aliases

abelian	C₂×C₃×C₄²×C₃₂
multiplicative	(ℤ/7760)ˣ, (ℤ/8320)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₆	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

3594 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃×C₄²×C₆₄, C₂×C₃×C₄×C₈×C₃₂, C₃×C₄³×C₃₂, …
maximal normal subgroup of	C₂×C₃×C₄²×C₆₄, C₂×C₃×C₄×C₈×C₃₂, C₃×C₄³×C₃₂, …
maximal quotient of	C₂×C₃×C₄²×C₆₄, C₂×C₃×C₄×C₈×C₃₂, C₃×C₄³×C₃₂, …
automorphism group of	C₅×C₁₆×C₉₇, C₅×C₁₃×C₁₂₈
outer automorphism group of	C₅×C₁₆×C₉₇, C₅×C₁₃×C₁₂₈

GAP

Past the SmallGroups library coverage — no SmallGroups id.