#65480

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×Q₁₆ ⋊ C₂
Sylow	C₅ ⋊ #495

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₅×Q₁₆	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₄₀	C₂
hypercenter	C₂×Q₁₆	D₅
hypocenter	C₅	#495
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598808, #10600009, #10600010, …
maximal normal subgroup of	#10598808, #10600009, #10600010, …
maximal quotient of	#10601019, #10601044, #10601050, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 818);