#65426

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #470

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#470
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

110 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597973, #10597975, #10598027, …
maximal normal subgroup of	#10597973, #10597975, #10598027, …
maximal quotient of	#10600610, #10600613, #10600617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 764);