C₅×#2063 #10603571

order	640
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#2063

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#2063, #12127844, #12127853, …
maximal subgroup of	C₂×C₅×#2063, C₅×#23601, C₅×#23602, …
maximal normal subgroup of	C₂×C₅×#2063, C₅×#23601, C₅×#23602, …
maximal quotient of	C₅×#10587, C₅×#10589, C₅×#10591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5653);