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C₉×ΓL(1,𝔽₈) #4426

order	189
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×D₃×AGL(1,𝔽₇) (order 756) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

C₉×ΓL(1,𝔽₈)

Splittings

Fitting	C₇×C₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃×C₉
Frattini	C₃	C₃×ΓL(1,𝔽₈)
Fitting	C₇×C₉	C₃
socle, cosocle	C₃×C₇	C₃²
prime radical	C₃×ΓL(1,𝔽₈)	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₉

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3,4)(5,7,6);(1,2,3,5,4,6,7)(8,9,11,10,12,14,13,15,16)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₉×AGL(1,𝔽₇), D₉×ΓL(1,𝔽₈), ΓL(1,𝔽₈)⋊D₉, …
maximal normal subgroup of	C₉×AGL(1,𝔽₇), D₉×ΓL(1,𝔽₈), ΓL(1,𝔽₈)⋊D₉, …
maximal quotient of	C₂×C₉×ΓL(1,𝔽₈), ΓL(1,𝔽₈)×C₂₇, #10588029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(189, 3);