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C₂×C₃³×C₄×C₆₄ #72192000

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 184025088
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₃³×C₄

Aliases

abelian	C₂×C₃³×C₄×C₆₄
multiplicative	(ℤ/17563)ˣ, (ℤ/22581)ˣ, (ℤ/35126)ˣ, (ℤ/45162)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃₂	C₂³×C₃³
socle, prime radical	C₂³×C₃³	C₂×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

4536 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	C₂×C₃³×C₄×C₁₂₈, C₂×C₃³×C₈×C₆₄, C₃³×C₄²×C₆₄, …
maximal normal subgroup of	C₂×C₃³×C₄×C₁₂₈, C₂×C₃³×C₈×C₆₄, C₃³×C₄²×C₆₄, …
maximal quotient of	C₂×C₃³×C₄×C₁₂₈, C₂×C₃³×C₈×C₆₄, C₃³×C₄²×C₆₄, …
automorphism group of	C₇×C₁₃×C₁₉₃
outer automorphism group of	C₇×C₁₃×C₁₉₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.