Finite Group Databasebeta

← Prev (#66173) Next (#66175) →

C₂×#4099 #66174

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

Splittings

Fitting	C₂⁵×C₅ ⋊ C₂
Sylow	C₅ ⋊ C₂×#120

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂⁵×C₅	C₂
socle	C₂³×C₅	C₂³
hypercenter	C₂⁵	D₅
hypocenter	C₅	C₂×#120
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁵

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600289, #10600292, #10600293, …
maximal normal subgroup of	#10600289, #10600292, #10600293, …
maximal quotient of	C₂×#65499, C₂×#65502, C₄×#4099, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1512);