#65009

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #433

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₅×C₂²⋊C₈	C₂
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂²⋊C₈	D₅
hypocenter	C₅	#433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#65009, #10610315, #10610335, …
maximal normal subgroup of	C₂×#65009, #10610315, #10610335, …
maximal quotient of	#10599391, #10599397, #10599399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 347);