C₅×#1616 #10603124

order	640
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#1616

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
prime radical	C₅×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

246 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12011536, #12011543, #12011550, …
maximal subgroup of	C₅×#12127, C₅×#12128, C₅×#12129, …
maximal normal subgroup of	C₅×#12127, C₅×#12128, C₅×#12129, …
maximal quotient of	C₅×#7791, C₅×#7795, C₅×#7808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5206);