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C₂²×C₃×C₄×C₁₆² #72190671

order	12288
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 618475290624
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁷×C₄²×C₁₆

Aliases

abelian	C₂²×C₃×C₄×C₁₆²
multiplicative	(ℤ/38080)ˣ, (ℤ/42432)ˣ, (ℤ/48960)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈²	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂×C₈²
LCS	G ▷ C₁
UCS	C₁ ◁ G

63030 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12288 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₃×C₄×C₁₆×C₃₂, C₂²×C₃×C₈×C₁₆², C₂×C₃×C₄²×C₁₆², …
maximal normal subgroup of	C₂²×C₃×C₄×C₁₆×C₃₂, C₂²×C₃×C₈×C₁₆², C₂×C₃×C₄²×C₁₆², …
maximal quotient of	C₂²×C₃×C₄×C₁₆×C₃₂, C₂²×C₃×C₈×C₁₆², C₂×C₃×C₄²×C₁₆², …

GAP

Past the SmallGroups library coverage — no SmallGroups id.