#65468

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 ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	C₂×#686
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₂×C₅×Q₁₆	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4050	C₂
hypercenter	C₂×Q₁₆	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

82 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)(6,9,12,22)(7,14,16,30)(8,17,11,20)(10,24,26,37)(13,28,27,29)(15,31,32,33)(18,34,25,21)(19,35,36,23);(6,8,12,11)(7,13,16,27)(9,20,22,17)(10,18,26,25)(14,29,30,28)(15,23,32,35)(19,31,36,33)(21,37,34,24);(2,3)(4,5)(6,7)(8,13)(9,19)(10,23)(11,27)(12,16)(14,21)(15,25)(17,33)(18,32)(20,31)(22,36)(24,28)(26,35)(29,37)(30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600057, #10600065, #10600066, …
maximal normal subgroup of	#10600057, #10600065, #10600066, …
maximal quotient of	#10601389, #10601391, #10601392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 806);