#65408

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #424

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×M₁₆	C₂
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

114 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)(10,12,13,19,14,18,20,24)(11,16,17,23,15,21,22,25);(11,15)(12,18)(17,22)(19,24);(2,3)(4,5)(6,7,8,9)(10,11)(12,16)(13,17)(14,15)(18,21)(19,23)(20,22)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598024, #10598782, #10599491, …
maximal normal subgroup of	#10598024, #10598782, #10599491, …
maximal quotient of	#10600625, #10600629, #10600702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 746);