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C₂×C₅×C₃⋊D₃×C₃₁ #19150179

order	5580
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₃₁ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₅×C₃⋊D₃×C₃₁
unambiguous semidirect product	(C₂×C₃)⋊(C₅×D₃×C₃₁)

Splittings

Fitting, socle	C₂×C₃²×C₅×C₃₁ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂²×C₅×C₃₁
Hall	C₃²×C₅×C₃₁ ⋊ C₂²
Hall	C₃²×C₃₁ ⋊ C₂²×C₅
Hall	C₃²×C₅ ⋊ C₂²×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅×C₃₁	C₃⋊D₃
commutator, hypocenter	C₃²	C₂²×C₅×C₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅×C₃₁	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅×C₃₁

120 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1860 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃⋊D₃×C₃₁⋊C₂₅, C₂×C₃⋊D₃×C₂₅×C₃₁, C₂×C₅×C₃⋊D₃×ΓL(1,𝔽₃₂), …
maximal normal subgroup of	C₂×C₃⋊D₃×C₃₁⋊C₂₅, C₂×C₃⋊D₃×C₂₅×C₃₁, C₂×C₅×C₃⋊D₃×ΓL(1,𝔽₃₂), …
maximal quotient of	C₂×C₃⋊D₃×C₂₅×C₃₁, C₅×D₅×C₃⋊D₃×C₃₁, C₂×C₅²×C₃⋊D₃×C₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5580, 168);