C₅×#1723 #10603231

order	640
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#1723

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

176 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 640

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

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1723, #12087102, #12087116, …
maximal subgroup of	C₂×C₅×#1723, C₅×#19618, C₅×#19619, …
maximal normal subgroup of	C₂×C₅×#1723, C₅×#19618, C₅×#19619, …
maximal quotient of	C₅×#8820, C₅×#9833, C₅×#9834, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5313);