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C₄×C₁₉×ΓL(1,𝔽₃₂) #26044508

order	11780
isoclinism	isoclinic class of ΓL(1,𝔽₃₂)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×C₁₉×ΓL(1,𝔽₃₂)
unambiguous semidirect product	(C₄×C₃₁)⋊(C₅×C₁₉)
unambiguous semidirect product	(C₄×C₁₉×C₃₁)⋊C₅

Splittings

Fitting, Hall	C₄×C₁₉×C₃₁ ⋊ C₅
derived, Sylow	C₃₁ ⋊ C₄×C₅×C₁₉
Hall	C₁₉×C₃₁ ⋊ C₄×C₅
Hall	C₄×C₃₁ ⋊ C₅×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₉	ΓL(1,𝔽₃₂)
commutator, hypocenter	C₃₁	C₄×C₅×C₁₉
Frattini	C₂	C₂×C₁₉×ΓL(1,𝔽₃₂)
Fitting	C₄×C₁₉×C₃₁	C₅
socle	C₂×C₁₉×C₃₁	C₂×C₅
prime radical	C₂×C₁₉×ΓL(1,𝔽₃₂)	C₂
cosocle	C₂×C₃₁	C₂×C₅×C₁₉
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₄×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

836 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×C₄×C₁₉×ΓL(1,𝔽₃₂), C₄×C₁₉×(C₃₁⋊C₃)⋊C₅, C₄×C₁₉⋊C₃×ΓL(1,𝔽₃₂), …
maximal normal subgroup of	C₃×C₄×C₁₉×ΓL(1,𝔽₃₂), C₄×C₁₉×(C₃₁⋊C₃)⋊C₅, C₄×C₁₉⋊C₃×ΓL(1,𝔽₃₂), …
maximal quotient of	C₃×C₄×C₁₉×ΓL(1,𝔽₃₂), Dic₃×C₁₉×ΓL(1,𝔽₃₂)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11780, 2);