#64730

order	320
isoclinism	isoclinic class of D₈₀
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₁₆ ⋊ C₂
Sylow	C₅ ⋊ #359

Subgroups

	subgroup	G/H
center	C₄	D₄₀
commutator	C₅×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	#102
prime radical	#4050	C₂
cosocle	C₂×C₅×C₈	C₂²
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#359
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,4,5,3)(6,8,9,17,21,31,36,37,11,19,23,33,10,18,22,32)(7,12,25,29,24,28,20,30,14,26,13,16,15,27,35,34);(2,3)(4,5)(6,7)(8,16)(9,20)(10,24)(11,14)(12,17)(13,22)(15,21)(18,34)(19,29)(23,35)(25,36)(26,33)(27,37)(28,32)(30,31)

Polycyclic presentation:

2^6,5|b^2=cd,c^2=ef,e^2=f,b^a=bc,c^a=ce,d^a=df,e^a=ef,g^a=g^4

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599860, #10599878, #10599919, …
maximal normal subgroup of	#10599860, #10599878, #10599919, …
maximal quotient of	#10597995, #10597996, #10598018, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 68);