#65193

order	320
isoclinism	isoclinic class of D₈₀
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₁₆ ⋊ C₂
Sylow	C₅ ⋊ #508

Subgroups

	subgroup	G/H
center	C₄	D₄₀
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	C₂×D₈
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#508
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 37 points:

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

Polycyclic presentation:

2^6,5|b^2=f,c^2=de,d^2=ef,e^2=f,c^a=cd,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	#10598071, #10598076, #10598089, …
maximal normal subgroup of	#10598071, #10598076, #10598089, …
maximal quotient of	#10599838, C₄×D₈₀, #10599852, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 531);