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C₂³×C₃×C₄²×C₂₇ #72189031

order	10368
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 21403533312
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁹×C₃×C₄

Aliases

abelian	C₂³×C₃×C₄²×C₂₇
multiplicative	(ℤ/30520)ˣ, (ℤ/34008)ˣ, (ℤ/36624)ˣ, (ℤ/39240)ˣ, (ℤ/42120)ˣ, (ℤ/45360)ˣ, (ℤ/45780)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₉	C₂⁵×C₃²
socle, prime radical	C₂⁵×C₃²	C₂²×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

21000 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10368 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×C₃×C₄×C₈×C₂₇, C₂²×C₃×C₄³×C₂₇, C₂⁴×C₃×C₄²×C₂₇, …
maximal normal subgroup of	C₂³×C₃×C₄×C₈×C₂₇, C₂²×C₃×C₄³×C₂₇, C₂⁴×C₃×C₄²×C₂₇, …
maximal quotient of	C₂³×C₃×C₄×C₈×C₂₇, C₂²×C₃×C₄³×C₂₇, C₂⁴×C₃×C₄²×C₂₇, …
automorphism group of	C₅×C₇×C₈×C₁₀₉
outer automorphism group of	C₅×C₇×C₈×C₁₀₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.