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C₂²×C₈×C₂₄₃ #72186708

order	7776
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 62208
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³

Aliases

abelian	C₂²×C₈×C₂₄₃
multiplicative	(ℤ/15584)ˣ, (ℤ/23328)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈₁	C₂³×C₃
socle, prime radical	C₂³×C₃	C₄×C₈₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

228 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

7776 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₁₆×C₂₄₃, C₂×C₄×C₈×C₂₄₃, C₂³×C₈×C₂₄₃, …
maximal normal subgroup of	C₂²×C₁₆×C₂₄₃, C₂×C₄×C₈×C₂₄₃, C₂³×C₈×C₂₄₃, …
maximal quotient of	C₂²×C₁₆×C₂₄₃, C₂×C₄×C₈×C₂₄₃, C₂³×C₈×C₂₄₃, …
automorphism group of	C₃₂×C₇₂₉
outer automorphism group of	C₃₂×C₇₂₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.