#64989

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅×C₈ ⋊ C₂
Sylow	C₅ ⋊ #443

Subgroups

	subgroup	G/H
center	C₈	D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle	C₂×C₅	C₄×D₄
prime radical	#4050	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#443
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

106 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:

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

Polycyclic presentation:

2^6,5|b^2=e,c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^4

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597932, #10597934, #10598059, …
maximal normal subgroup of	#10597932, #10597934, #10598059, …
maximal quotient of	C₈×D₄₀, #10598845, C₈×#677, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 327);