C₅×#1628 #10603136

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

Aliases

direct product

C₅×#1628

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₅×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

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

normal subgroups only

This group has 216 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) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1628, #12075063, #12075309, …
maximal subgroup of	C₂×C₅×#1628, C₅×#18494, C₅×#18520, …
maximal normal subgroup of	C₂×C₅×#1628, C₅×#18494, C₅×#18520, …
maximal quotient of	C₅×#7740, C₅×#7744, C₅×#7746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5218);