#65384

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×C₄⋊Q₈ ⋊ C₂
Sylow	C₅ ⋊ #456

Subgroups

	subgroup	G/H
center	C₂	#4083
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle	C₂×C₅	#120
prime radical	#4147	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#456
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597954, #10598041, #10599960, …
maximal normal subgroup of	#10597954, #10598041, #10599960, …
maximal quotient of	#10600366, #10600385, #10600386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 722);