Finite Group Databasebeta

← Prev (#65713) Next (#65715) →

C₂×#3990 #65714

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₅ ⋊ C₂×M₃₂

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₂×C₈
prime radical	C₂²×D₅	C₈
cosocle	#168	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(3,4,5,7,6)(8,10,12,19,13,11,18,20)(9,14,15,21,16,17,22,23);(4,5,6,7)(8,9,11,14,12,15,20,21,13,16,10,17,18,22,19,23);(1,2)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598621, #10598622, #10598630, …
maximal normal subgroup of	#10598621, #10598622, #10598630, …
maximal quotient of	C₄×#3990, #10603827, #10603828, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1052);