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C₅×AGL(1,𝔽₉) #67193

order	360
isoclinism	isoclinic class of AGL(1,𝔽₉)
automorphism	Aut(G) = C₄×AΓL(1,𝔽₉) (order 576) · Inn(G) = AGL(1,𝔽₉) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₅×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₅×AGL(1,𝔽₉)

Splittings

Fitting, socle, Hall	C₃²×C₅ ⋊ C₈
derived, Sylow	C₃² ⋊ C₅×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₅	AGL(1,𝔽₉)
commutator, hypocenter	C₃²	C₅×C₈
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₈
prime radical	C₅×C₃⋊D₃	C₄
cosocle	#157	C₂×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×AGL(1,𝔽₉), C₅×AΓL(1,𝔽₉), AGL(1,𝔽₉)⋊D₅, …
maximal normal subgroup of	D₅×AGL(1,𝔽₉), C₅×AΓL(1,𝔽₉), AGL(1,𝔽₉)⋊D₅, …
maximal quotient of	C₅×#3771, C₂×C₅×AGL(1,𝔽₉), C₅×#6249, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 123);