#65041

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#101 ⋊ C₂
Sylow	C₅ ⋊ #456

Subgroups

	subgroup	G/H
center	C₂	#4028
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#101	C₂
socle	C₂×C₅	#120
prime radical	#4147	C₂
hypercenter	#101	D₅
hypocenter	C₅	#456
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

142 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)(6,9,10,19,11,20,21,12)(7,13,14,18,15,8,16,17);(6,8,11,13)(7,12,15,19)(9,14,20,16)(10,18,21,17);(2,3)(4,5)(6,7)(8,12)(9,18)(10,16)(11,15)(13,19)(14,21)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598576, #10599954, #10599955, …
maximal normal subgroup of	#10598576, #10599954, #10599955, …
maximal quotient of	#10598925, #10598926, #10598927, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 379);