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C₃×C₁₆×C₂₅₁ #72190442

order	12048
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₄×C₁₂₅ (order 4000) · Inn(G) = C₁ · Out(G) = C₂³×C₄×C₁₂₅
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₀₄₈
abelian	C₃×C₁₆×C₂₅₁
multiplicative	(ℤ/12049)ˣ, (ℤ/24098)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃×C₂₅₁
socle, prime radical	C₂×C₃×C₂₅₁	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12048 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₀₄₈, D₂₄₀₉₆
maximal subgroup of	C₃×C₃₂×C₂₅₁, C₂×C₃×C₁₆×C₂₅₁, D₁₂₀₄₈, …
maximal normal subgroup of	C₃×C₃₂×C₂₅₁, C₂×C₃×C₁₆×C₂₅₁, D₁₂₀₄₈, …
maximal quotient of	C₃×C₃₂×C₂₅₁, C₂×C₃×C₁₆×C₂₅₁
automorphism group of	C₁₂₀₄₉
outer automorphism group of	C₁₂₀₄₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.