C₁₉×ΓL(1,𝔽₈)×D₁₇ #27214501

order	13566
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×ΓL(1,𝔽₈)×D₁₇
unambiguous semidirect product	C₁₇⋊(C₂×C₁₉×ΓL(1,𝔽₈))
unambiguous semidirect product	(C₇×D₁₇)⋊(C₃×C₁₉)

Splittings

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	ΓL(1,𝔽₈)×D₁₇
commutator, cosocle, hypocenter	C₇×C₁₇	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₁₉	C₂×C₃
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₁₉

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 2261

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,20,21)(22,24,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,20,22,21,23,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₉×ΓL(1,𝔽₈)⋊#595, ΓL(1,𝔽₈)⋊(C₁₉⋊#595), C₁₉×D₁₇×AGL(1,𝔽₇), …
maximal normal subgroup of	C₁₉×ΓL(1,𝔽₈)⋊#595, ΓL(1,𝔽₈)⋊(C₁₉⋊#595), C₁₉×D₁₇×AGL(1,𝔽₇), …
maximal quotient of	C₁₉×ΓL(1,𝔽₈)×Dic₁₇, C₂×C₁₉×ΓL(1,𝔽₈)×D₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13566, 26);

Fitting, socle, Hall	C₇×C₁₇×C₁₉ ⋊ C₂×C₃
derived, Hall	C₇×C₁₇ ⋊ C₂×C₃×C₁₉
Hall	C₁₇×C₁₉×ΓL(1,𝔽₈) ⋊ C₂
Hall	C₇×C₁₉×D₁₇ ⋊ C₃
Hall	C₁₇×ΓL(1,𝔽₈) ⋊ C₂×C₁₉
Hall	C₁₇×C₁₉ ⋊ C₂×ΓL(1,𝔽₈)
Hall	C₇×D₁₇ ⋊ C₃×C₁₉
Hall	C₇×C₁₉ ⋊ C₃×D₁₇
Sylow	C₁₇ ⋊ C₂×C₁₉×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ C₃×C₁₉×D₁₇

C19×ΓL(1,𝔽8)×D17 #27214501