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C₃×C₁₉×ΓL(1,𝔽₈)⋊Dic₇ #45691865

order	33516
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₁₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₁₉×ΓL(1,𝔽₈)⋊Dic₇
unambiguous semidirect product	ΓL(1,𝔽₈)⋊(C₃×C₁₉×Dic₇)

Splittings

derived, Sylow	C₇² ⋊ C₃²×C₄×C₁₉
Hall	C₃×C₇×C₁₉×ΓL(1,𝔽₈) ⋊ C₄
Hall	C₁₉×C₇⋊Dic₇ ⋊ C₃²
Hall	C₇²×C₁₉ ⋊ C₃²×C₄
Hall	C₃×C₇×ΓL(1,𝔽₈) ⋊ C₄×C₁₉
Hall	C₇⋊Dic₇ ⋊ C₃²×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇
commutator, hypocenter	C₇²	C₃²×C₄×C₁₉
Frattini	C₂	C₃×C₁₉×ΓL(1,𝔽₈)⋊D₇
Fitting, socle	C₂×C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₂×C₃×C₇×C₁₉×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇²	C₂×C₃²×C₁₉
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₁₉

168 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 4788

Conjugacy classes & character table

2508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 663);