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C₂²×C₄²×C₈×C₂₇ #72192048

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 43486543872
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

abelian	C₂²×C₄²×C₈×C₂₇
multiplicative	(ℤ/52320)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄×C₉	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂²×C₄×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

26376 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	C₂²×C₄²×C₁₆×C₂₇, C₂²×C₄×C₈²×C₂₇, C₂×C₄³×C₈×C₂₇, …
maximal normal subgroup of	C₂²×C₄²×C₁₆×C₂₇, C₂²×C₄×C₈²×C₂₇, C₂×C₄³×C₈×C₂₇, …
maximal quotient of	C₂²×C₄²×C₁₆×C₂₇, C₂²×C₄×C₈²×C₂₇, C₂×C₄³×C₈×C₂₇, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.