Finite Group Databasebeta

← Prev (#66272) Next (#66274) →

C₂²×#709 #66273

order	320
isoclinism	isoclinic class of Dic₁₀
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×#709

Splittings

Fitting	C₂³×C₄×C₅ ⋊ C₂
Sylow	C₅ ⋊ C₂²×#41

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁵
Frattini	C₂	C₂⁴×D₅
Fitting	C₂³×C₄×C₅	C₂
socle	C₂³×C₅	C₂³
hypercenter	C₂³×C₄	D₅
hypocenter	C₅	C₂²×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600247, #10600315, #10600330, …
maximal normal subgroup of	#10600247, #10600315, #10600330, …
maximal quotient of	C₂²×C₄×Dic₁₀, C₂×C₄×#709, C₂²×#4016, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1611);