Finite Group Databasebeta

← Prev (#10623644) Next (#10623646) →

C₉×C₄⋊AGL(1,𝔽₅) #10623645

order	720
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₉×C₄⋊AGL(1,𝔽₅)
unambiguous semidirect product	C₄⋊(C₉×AGL(1,𝔽₅))

Splittings

Fitting	C₄×C₅×C₉ ⋊ C₄
Hall	C₅×C₉ ⋊ C₄⋊C₄
Sylow	C₅ ⋊ C₉×C₄⋊C₄

Subgroups

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

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,14,13)(15,16,17,18);(11,12,13,14)(16,18)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13083914, #13083915, #13083916, …
maximal normal subgroup of	#13083914, #13083915, #13083916, …
maximal quotient of	C₉×#3993, C₉×#3994, C₉×#3995, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 114);