C₅×#1644 #10603152

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

Aliases

direct product

C₅×#1644

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1644, #12074989, #12075296, …
maximal subgroup of	C₂×C₅×#1644, C₅×#18485, C₅×#18519, …
maximal normal subgroup of	C₂×C₅×#1644, C₅×#18485, C₅×#18519, …
maximal quotient of	C₅×#7774, C₅×#7777, C₅×#7782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5234);