Finite Group Databasebeta

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C₂×C₃×C₄×C₈²×C₉ #72192040

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 5435817984
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³×C₃×C₄²×C₈

Aliases

abelian	C₂×C₃×C₄×C₈²×C₉
multiplicative	(ℤ/30368)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃×C₄²	C₂⁴×C₃²
socle, prime radical	C₂⁴×C₃²	C₂×C₃×C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

19220 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×C₃×C₄×C₈×C₉×C₁₆, C₂×C₃×C₈³×C₉, C₃×C₄²×C₈²×C₉, …
maximal normal subgroup of	C₂×C₃×C₄×C₈×C₉×C₁₆, C₂×C₃×C₈³×C₉, C₃×C₄²×C₈²×C₉, …
maximal quotient of	C₂×C₃×C₄×C₈×C₉×C₁₆, C₂×C₃×C₈³×C₉, C₃×C₄²×C₈²×C₉, …
automorphism group of	C₁₃×C₃₂×C₇₃
outer automorphism group of	C₁₃×C₃₂×C₇₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.