#65310

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #477

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#477
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599039, #10599040, #10599046, …
maximal normal subgroup of	#10599039, #10599040, #10599046, …
maximal quotient of	#10600254, #10600259, #10600282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 648);