Finite Group Databasebeta

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C₇×C₃₇×D₇⋊C₉ #45657307

order	32634
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₉×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₃₇×D₇⋊C₉
unambiguous semidirect product	D₇⋊(C₇×C₉×C₃₇)
unambiguous semidirect product	(C₇⋊C₉)⋊(C₂×C₇×C₃₇)

Splittings

derived	C₇ ⋊ C₂×C₇×C₉×C₃₇
Hall	C₇×C₃₇×C₇⋊C₉ ⋊ C₂
Hall	C₇×D₇×C₃₇ ⋊ C₉
Hall	C₇²×C₃₇ ⋊ C₂×C₉
Hall	C₇×C₇⋊C₉ ⋊ C₂×C₃₇
Hall	C₇×D₇ ⋊ C₉×C₃₇
Sylow	C₇² ⋊ C₂×C₉×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇×C₃₇	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₇×C₉×C₃₇
Frattini	C₃	C₇×C₃₇×AGL(1,𝔽₇)
Fitting, socle	C₃×C₇²×C₃₇	C₂×C₃
prime radical	C₃×C₇×D₇×C₃₇	C₃
cosocle	C₃×C₇	C₂×C₃×C₇×C₃₇
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₇×C₃₇

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 4662

Conjugacy classes & character table

5439 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32634, 87);