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C₂×C₉⋊AGL(1,𝔽₅) #67114

order	360
isoclinism	isoclinic class of C₉⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₉⋊AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊Dic₉
unambiguous semidirect product	(C₂×C₉)⋊AGL(1,𝔽₅)

Splittings

Fitting	C₂×C₅×C₉ ⋊ C₄
derived, Hall	C₅×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₂×Dic₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₉⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₉	C₂×C₄
Frattini	C₃	C₂×ΓL(1,𝔽₁₆)
Fitting	C₂×C₅×C₉	C₄
socle	C₂×C₃×C₅	Dic₃
prime radical	C₂×C₃×D₅	D₃
cosocle	C₉×D₅	C₂²
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 90

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

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×Dic₉, Dic₉⋊AGL(1,𝔽₅), (C₂×D₉)⋊AGL(1,𝔽₅), …
maximal normal subgroup of	AGL(1,𝔽₅)×Dic₉, Dic₉⋊AGL(1,𝔽₅), (C₂×D₉)⋊AGL(1,𝔽₅), …
maximal quotient of	D₅⋊(C₉⋊C₈), #10623650, C₄×C₉⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 44);