Finite Group Databasebeta

← Prev (#65201) Next (#65203) →

D₅×SD₃₂ #65202

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₅×SD₃₂

Splittings

Fitting	C₅×SD₃₂ ⋊ C₂
Sylow	C₅ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×SD₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	D₅×D₈	C₂
hypercenter	SD₃₂	D₅
hypocenter	C₅	C₂×SD₃₂
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(17,18,20,21,19);(1,2,5,9)(3,6,12,15)(4,13,11,8)(7,16,14,10);(18,19)(20,21)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×SD₃₂, #10604127, C₂×D₅×SD₃₂, …
maximal normal subgroup of	AGL(1,𝔽₅)×SD₃₂, #10604127, C₂×D₅×SD₃₂, …
maximal quotient of	#10599968, #10599972, #10599976, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 540);