#65057

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#102	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×C₅×D₄	C₂²
hypercenter	#102	D₅
hypocenter	C₅	#488
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 160

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

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

Polycyclic presentation:

2^6,5|a^2=e,c^2=e,d^2=f,b^a=bef,c^a=cef,g^a=g^4,c^b=cd,d^b=df,d^c=df

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#65057, #10610468, #10610490, …
maximal normal subgroup of	C₂×#65057, #10610468, #10610490, …
maximal quotient of	#10599597, #10599605, #10599638, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 395);