#65478

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×Q₁₆ ⋊ C₂
Sylow	C₅ ⋊ #470

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	C₂²×D₅	D₄
hypercenter	C₂×Q₁₆	D₅
hypocenter	C₅	#470
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598811, #10598812, #10600060, …
maximal normal subgroup of	#10598811, #10598812, #10600060, …
maximal quotient of	#10601036, #10601040, #10601049, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 816);