#65701

order	320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #405

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Fitting	C₅×C₄⋊C₄	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,5,4)(6,7,8,9)(12,18)(15,21)(19,24)(22,25);(10,12,14,18)(11,15,17,21)(13,19,20,24)(16,22,23,25);(2,3,4,5)(10,11,13,16,14,17,20,23)(12,15,19,22,18,21,24,25)

Polycyclic presentation:

2^6,5|a^2=d,b^2=f,c^2=e,d^2=f,g^a=g^2,c^b=cf,g^d=g^4

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10603959, #10603960, #10603961, …
maximal normal subgroup of	#10603959, #10603960, #10603961, …
maximal quotient of	#10603773, #10603774, #10603775, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1039);