#64726

order	320
isoclinism	isoclinic class of D₈₀
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #368

Subgroups

	subgroup	G/H
center	C₄	D₄₀
commutator	C₅×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	#107
prime radical	C₂²×C₅	Q₁₆
cosocle	C₂×C₅×C₈	C₂²
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#368
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,4,5,3)(6,8,18,34,11,21,25,35,12,22,23,27,9,19,10,20)(7,13,24,31,14,28,30,33,16,29,37,36,26,32,15,17);(2,3)(4,5)(6,7,9,14,12,16,11,26)(8,17,19,31,22,33,21,36)(10,24,23,30,25,37,18,15)(13,27,28,35,29,34,32,20)

Polycyclic presentation:

2^6,5|a^2=c,b^2=de,c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^4

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598070, #10598072, #10598084, …
maximal normal subgroup of	#10598070, #10598072, #10598084, …
maximal quotient of	#10597991, #10597992, #10598299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 64);