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C₂×C₄²×C₁₆×C₂₇ #72192027

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 226492416
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³×C₄³

Aliases

abelian	C₂×C₄²×C₁₆×C₂₇
multiplicative	(ℤ/27795)ˣ, (ℤ/29648)ˣ, (ℤ/34640)ˣ, (ℤ/34880)ˣ, (ℤ/37060)ˣ, (ℤ/55590)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈×C₉	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

5640 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

13824 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄²×C₂₇×C₃₂, C₂×C₄×C₈×C₁₆×C₂₇, C₄³×C₁₆×C₂₇, …
maximal normal subgroup of	C₂×C₄²×C₂₇×C₃₂, C₂×C₄×C₈×C₁₆×C₂₇, C₄³×C₁₆×C₂₇, …
maximal quotient of	C₂×C₄²×C₂₇×C₃₂, C₂×C₄×C₈×C₁₆×C₂₇, C₄³×C₁₆×C₂₇, …
automorphism group of	C₁₆×C₁₇×C₁₀₉
outer automorphism group of	C₁₆×C₁₇×C₁₀₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.