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C₂³×C₃×C₉×C₆₄ #72192001

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 37158912
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

abelian	C₂³×C₃×C₉×C₆₄
multiplicative	(ℤ/29336)ˣ, (ℤ/32312)ˣ, (ℤ/34048)ˣ, (ℤ/41544)ˣ, (ℤ/41688)ˣ, (ℤ/43776)ˣ, (ℤ/44004)ˣ, (ℤ/48384)ˣ, (ℤ/48468)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₃₂	C₂⁴×C₃²
socle, prime radical	C₂⁴×C₃²	C₃×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

3220 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	C₂³×C₃×C₉×C₁₂₈, C₂²×C₃×C₄×C₉×C₆₄, C₂⁴×C₃×C₉×C₆₄
maximal normal subgroup of	C₂³×C₃×C₉×C₁₂₈, C₂²×C₃×C₄×C₉×C₆₄, C₂⁴×C₃×C₉×C₆₄
maximal quotient of	C₂³×C₃×C₉×C₁₂₈, C₂²×C₃×C₄×C₉×C₆₄, C₂⁴×C₃×C₉×C₆₄
automorphism group of	C₈×C₁₉×C₁₉₃, C₇×C₈×C₅₇₇
outer automorphism group of	C₈×C₁₉×C₁₉₃, C₇×C₈×C₅₇₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.