Finite Group Databasebeta

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C₂²×C₄×C₂₇×C₂₉ #72190875

order	12528
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 96768
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³

Aliases

abelian	C₂²×C₄×C₂₇×C₂₉
multiplicative	(ℤ/19293)ˣ, (ℤ/23505)ˣ, (ℤ/23895)ˣ, (ℤ/25072)ˣ, (ℤ/25724)ˣ, (ℤ/31340)ˣ, (ℤ/38586)ˣ, (ℤ/47010)ˣ, (ℤ/47790)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉	C₂³×C₃×C₂₉
socle, prime radical	C₂³×C₃×C₂₉	C₂×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

216 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12528 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₈×C₂₇×C₂₉, C₂×C₄²×C₂₇×C₂₉, C₂³×C₄×C₂₇×C₂₉
maximal normal subgroup of	C₂²×C₈×C₂₇×C₂₉, C₂×C₄²×C₂₇×C₂₉, C₂³×C₄×C₂₇×C₂₉
maximal quotient of	C₂²×C₈×C₂₇×C₂₉, C₂×C₄²×C₂₇×C₂₉, C₂³×C₄×C₂₇×C₂₉
automorphism group of	C₅×C₅₉×C₈₁
outer automorphism group of	C₅×C₅₉×C₈₁

GAP

Past the SmallGroups library coverage — no SmallGroups id.