C₅×#1637 #10603145

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

Aliases

direct product

C₅×#1637

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

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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)(6,7,9,12)(8,10,13,11)(14,15,20,24)(16,25,17,28)(18,23,19,22)(21,27,29,26);(7,10)(8,13)(11,12)(14,16,20,17)(15,21,24,29)(18,27,19,26)(22,28,23,25);(14,17,20,16)(15,21,24,29)(18,26,19,27)(22,28,23,25)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1637, #12075069, #12075264, …
maximal subgroup of	C₂×C₅×#1637, C₅×#18494, C₅×#18517, …
maximal normal subgroup of	C₂×C₅×#1637, C₅×#18494, C₅×#18517, …
maximal quotient of	C₅×#7764, C₅×#7766, C₅×#7772, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5227);