C₅×#1643 #10603151

order	640
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#1643

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₂×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 320

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

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1643, #12074988, #12075281, …
maximal subgroup of	C₂×C₅×#1643, C₅×#18485, C₅×#18518, …
maximal normal subgroup of	C₂×C₅×#1643, C₅×#18485, C₅×#18518, …
maximal quotient of	C₅×#7753, C₅×#7755, C₅×#7767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5233);