#65139

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×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄⋊C₈	D₅
hypocenter	C₅	#477
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598981, #10598982, #10598995, …
maximal normal subgroup of	#10598981, #10598982, #10598995, …
maximal quotient of	#10599417, #10599425, #10599436, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 477);