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D₇×ΓL(1,𝔽₈)×C₃₇ #24925979

order	10878
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

D₇×ΓL(1,𝔽₈)×C₃₇

Splittings

Fitting, socle, Hall	C₇²×C₃₇ ⋊ C₂×C₃
derived, Sylow	C₇² ⋊ C₂×C₃×C₃₇
Hall	C₇×ΓL(1,𝔽₈)×C₃₇ ⋊ C₂
Hall	C₇×D₇×C₃₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₃₇
Hall	C₇×D₇ ⋊ C₃×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₃₇
Frattini	C₁	G
Fitting, socle	C₇²×C₃₇	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃₇

34 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

925 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×C₃₇×AGL(1,𝔽₇), D₇×ΓL(1,𝔽₈)⋊D₃₇, D₇×ΓL(1,𝔽₈)×D₃₇, …
maximal normal subgroup of	D₇×C₃₇×AGL(1,𝔽₇), D₇×ΓL(1,𝔽₈)⋊D₃₇, D₇×ΓL(1,𝔽₈)×D₃₇, …
maximal quotient of	ΓL(1,𝔽₈)×Dic₇×C₃₇, C₂×D₇×ΓL(1,𝔽₈)×C₃₇, D₇×C₃₇×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10878, 71);