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C₂×C₄²×C₈×C₂₇ #72186010

order	6912
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 113246208
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³×C₄³

Aliases

abelian	C₂×C₄²×C₈×C₂₇
multiplicative	(ℤ/17440)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄×C₉	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₄×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

4092 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

6912 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄²×C₁₆×C₂₇, C₂×C₄×C₈²×C₂₇, C₄³×C₈×C₂₇, …
maximal normal subgroup of	C₂×C₄²×C₁₆×C₂₇, C₂×C₄×C₈²×C₂₇, C₄³×C₈×C₂₇, …
maximal quotient of	C₂×C₄²×C₁₆×C₂₇, C₂×C₄×C₈²×C₂₇, C₄³×C₈×C₂₇, …
automorphism group of	C₅×C₃₂×C₁₀₉
outer automorphism group of	C₅×C₃₂×C₁₀₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.