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C₂²×C₄×C₈×C₂₇ #72183128

order	3456
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 3538944
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

abelian	C₂²×C₄×C₈×C₂₇
multiplicative	(ℤ/10464)ˣ, (ℤ/12960)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄×C₉	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂×C₄×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

1520 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3456 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₄×C₁₆×C₂₇, C₂²×C₈²×C₂₇, C₂×C₄²×C₈×C₂₇, …
maximal normal subgroup of	C₂²×C₄×C₁₆×C₂₇, C₂²×C₈²×C₂₇, C₂×C₄²×C₈×C₂₇, …
maximal quotient of	C₂²×C₄×C₁₆×C₂₇, C₂²×C₈²×C₂₇, C₂×C₄²×C₈×C₂₇, …
automorphism group of	C₃×C₃₂×C₁₀₉, C₅×C₃₂×C₈₁
outer automorphism group of	C₃×C₃₂×C₁₀₉, C₅×C₃₂×C₈₁

GAP

Past the SmallGroups library coverage — no SmallGroups id.