#65088

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#103	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#103	D₅
hypocenter	C₅	#479
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

2^6,5|a^2=d,b^2=f,c^2=df,e^2=f,c^a=cdf,g^a=g^4,c^b=ce,e^b=ef,e^c=ef

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#65088, #10610510, #10610538, …
maximal normal subgroup of	C₂×#65088, #10610510, #10610538, …
maximal quotient of	#10599590, #10599595, #10599600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 426);