C₅×#2097 #10603605

order	640
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#2097

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

238 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 33
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5)(10,11,13,15)(12,16,17,14)(18,19,21,25,22,26,30,23)(20,27,28,32,29,33,31,24);(6,7)(8,9)(10,12,13,17)(11,14,15,16)(19,23)(21,30)(24,27)(25,26)(28,31)(32,33);(7,9)(18,20,22,29)(19,24,26,32)(21,31,30,28)(23,27,25,33)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#2097, #12129099, #12129114, …
maximal subgroup of	C₂×C₅×#2097, C₅×#23730, C₅×#23732, …
maximal normal subgroup of	C₂×C₅×#2097, C₅×#23730, C₅×#23732, …
maximal quotient of	C₅×#10664, C₅×#10669, C₅×#10675, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5687);