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Dic₃×C₁₉×ΓL(1,𝔽₈) #18012186

order	4788
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×C₁₉×ΓL(1,𝔽₈)
unambiguous semidirect product	C₃⋊(C₄×C₁₉×ΓL(1,𝔽₈))
unambiguous semidirect product	(C₇×Dic₃)⋊(C₃×C₁₉)

Splittings

derived	C₃×C₇ ⋊ C₃×C₄×C₁₉
Hall	C₃×C₁₉×ΓL(1,𝔽₈) ⋊ C₄
Hall	C₇×C₁₉ ⋊ C₃×Dic₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₄×C₁₉
Sylow	C₇ ⋊ C₃×Dic₃×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₃×C₄×C₁₉
Frattini	C₂	D₃×C₁₉×ΓL(1,𝔽₈)
Fitting, socle	C₂×C₃×C₇×C₁₉	C₂×C₃
prime radical	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₃×C₇	C₂×C₃×C₁₉
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₁₉

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 1596

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×Dic₃×C₁₉×ΓL(1,𝔽₈), C₁₉×ΓL(1,𝔽₈)×Dic₁₅, C₁₉×ΓL(1,𝔽₈)×ΓL(1,𝔽₁₆), …
maximal normal subgroup of	C₅×Dic₃×C₁₉×ΓL(1,𝔽₈), C₇×Dic₃×C₁₉×ΓL(1,𝔽₈)
maximal quotient of	C₅×Dic₃×C₁₉×ΓL(1,𝔽₈), C₁₉×ΓL(1,𝔽₈)×Dic₁₅, C₁₉×ΓL(1,𝔽₈)×ΓL(1,𝔽₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 102);