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C₁₉×(C₇⋊C₉)⋊D₇ #30660593

order	16758
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×(C₇⋊C₉)⋊D₇
unambiguous semidirect product	(C₇⋊C₉)⋊(D₇×C₁₉)

Splittings

derived, Sylow	C₇² ⋊ C₂×C₉×C₁₉
Hall	C₇×C₁₉×C₇⋊C₉ ⋊ C₂
Hall	C₁₉×C₇⋊D₇ ⋊ C₉
Hall	C₇²×C₁₉ ⋊ C₂×C₉
Hall	C₇×C₇⋊C₉ ⋊ C₂×C₁₉
Hall	C₇⋊D₇ ⋊ C₉×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇
commutator, hypocenter	C₇²	C₂×C₉×C₁₉
Frattini	C₃	C₁₉×ΓL(1,𝔽₈)⋊D₇
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₃×C₁₉×C₇⋊D₇	C₃
cosocle	C₃×C₇²	C₂×C₃×C₁₉
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₁₉

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1197

Conjugacy classes & character table

1254 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×C₁₉×D₇⋊C₉, ((C₇⋊C₉)⋊D₇)⋊D₁₉, D₁₉×(C₇⋊C₉)⋊D₇, …
maximal normal subgroup of	D₇×C₁₉×D₇⋊C₉, ((C₇⋊C₉)⋊D₇)⋊D₁₉, D₁₉×(C₇⋊C₉)⋊D₇, …
maximal quotient of	C₁₉×(C₇⋊C₉)⋊Dic₇, C₂×C₁₉×(C₇⋊C₉)⋊D₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16758, 120);