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C₃×C₄²×C₂₂₃ #72189255

order	10704
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 42624
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

abelian	C₃×C₄²×C₂₂₃
multiplicative	(ℤ/13385)ˣ, (ℤ/26770)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃×C₂₂₃
socle, prime radical	C₂²×C₃×C₂₂₃	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10704 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₄×C₈×C₂₂₃, C₂×C₃×C₄²×C₂₂₃, C₄²×C₉×C₂₂₃, …
maximal normal subgroup of	C₃×C₄×C₈×C₂₂₃, C₂×C₃×C₄²×C₂₂₃, C₄²×C₉×C₂₂₃, …
maximal quotient of	C₃×C₄×C₈×C₂₂₃, C₂×C₃×C₄²×C₂₂₃, C₄²×C₉×C₂₂₃, …
automorphism group of	C₅×C₂₆₇₇
outer automorphism group of	C₅×C₂₆₇₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.