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C₂×(C₇⋊C₉)⋊D₁₉ #18012121

order	4788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×(C₇⋊C₉)⋊D₁₉
unambiguous semidirect product	(C₂×C₇⋊C₉)⋊D₁₉

Splittings

derived, Hall	C₇×C₁₉ ⋊ C₂²×C₉
Hall	C₁₉×C₇⋊C₉ ⋊ C₂²
Hall	C₂×D₁₃₃ ⋊ C₉
Hall	C₇⋊C₉ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×D₇⋊C₉
Sylow	C₇ ⋊ C₂×C₉×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)⋊D₁₉
commutator, hypocenter	C₇×C₁₉	C₂²×C₉
Frattini	C₃	C₂×ΓL(1,𝔽₈)⋊D₁₉
Fitting, socle	C₂×C₃×C₇×C₁₉	C₂×C₃
prime radical	C₂×C₃×D₁₃₃	C₃
cosocle	C₃×C₇×C₁₉	C₂²×C₃
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂×C₃

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 2394

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₅×(C₇⋊C₉)⋊D₁₉, C₂×C₅×(C₇⋊C₉)⋊D₁₉, C₂×C₅⋊((C₇⋊C₉)⋊D₁₉), …
maximal normal subgroup of	C₂×C₅×(C₇⋊C₉)⋊D₁₉, C₂×C₇×(C₇⋊C₉)⋊D₁₉
maximal quotient of	D₅×(C₇⋊C₉)⋊D₁₉, C₂×C₅×(C₇⋊C₉)⋊D₁₉, C₂×C₅⋊((C₇⋊C₉)⋊D₁₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 37);