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D₇×ΓL(1,𝔽₈)×C₁₉⋊C₃ #30660721

order	16758
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

D₇×ΓL(1,𝔽₈)×C₁₉⋊C₃

Splittings

Fitting, socle, derived, Hall	C₇²×C₁₉ ⋊ C₂×C₃²
Hall	C₇×ΓL(1,𝔽₈)×C₁₉⋊C₃ ⋊ C₂
Hall	C₇×D₇×C₁₉ ⋊ C₃²
Hall	C₇×D₇ ⋊ C₃×C₁₉⋊C₃
Sylow	C₇² ⋊ C₂×C₃×C₁₉⋊C₃
Sylow	C₁₉ ⋊ C₃×D₇×ΓL(1,𝔽₈)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₉	C₂×C₃²
LCS	G ▷ C₇²×C₁₉
UCS	C₁

100 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 798

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×ΓL(1,𝔽₈)⋊(D₁₉⋊C₃), D₇×AGL(1,𝔽₇)×C₁₉⋊C₃, D₇×ΓL(1,𝔽₈)×D₁₉⋊C₃, …
maximal normal subgroup of	D₇×ΓL(1,𝔽₈)⋊(D₁₉⋊C₃), D₇×AGL(1,𝔽₇)×C₁₉⋊C₃, D₇×ΓL(1,𝔽₈)×D₁₉⋊C₃, …
maximal quotient of	ΓL(1,𝔽₈)×Dic₇×C₁₉⋊C₃, C₂×D₇×ΓL(1,𝔽₈)×C₁₉⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16758, 248);