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C₅²×D₇×ΓL(1,𝔽₈) #20301259

order	7350
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅² · Schur multiplier (H₂) = C₅

Aliases

C₅²×D₇×ΓL(1,𝔽₈)

Splittings

Fitting, socle, Hall	C₅²×C₇² ⋊ C₂×C₃
derived, Sylow	C₇² ⋊ C₂×C₃×C₅²
Hall	C₅²×C₇×ΓL(1,𝔽₈) ⋊ C₂
Hall	C₅²×C₇×D₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₅²
Hall	C₇×D₇ ⋊ C₃×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₅²	D₇×ΓL(1,𝔽₈)
commutator, hypocenter	C₇²	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₅²×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₅²

136 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

625 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅²×D₇×AGL(1,𝔽₇), C₅×D₇×C₅⋊AGL(1,𝔽₇), C₅×D₅×D₇×ΓL(1,𝔽₈), …
maximal normal subgroup of	C₅²×D₇×AGL(1,𝔽₇), C₅×D₇×C₅⋊AGL(1,𝔽₇), C₅×D₅×D₇×ΓL(1,𝔽₈), …
maximal quotient of	C₅²×ΓL(1,𝔽₈)×Dic₇, C₂×C₅²×D₇×ΓL(1,𝔽₈), C₅²×D₇×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 82);