C₅×#1653 #10603161

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

Aliases

direct product

C₅×#1653

Subgroups

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

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

normal subgroups only

This group has 194 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) = 640

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1653, #12075146, #12075287, …
maximal subgroup of	C₂×C₅×#1653, C₅×#18503, C₅×#18518, …
maximal normal subgroup of	C₂×C₅×#1653, C₅×#18503, C₅×#18518, …
maximal quotient of	C₅×#7760, C₅×#7761, C₅×#7767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5243);