C₅×#2801 #10616396

order	640
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#2801

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

420 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 176 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 — 4 generators on 29 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12307324, #12307429, #12309076, …
maximal subgroup of	C₅×#36766, C₅×#36769, C₅×#36826, …
maximal normal subgroup of	C₅×#36766, C₅×#36769, C₅×#36826, …
maximal quotient of	C₅×#14434, C₅×#14459, C₅×#14504, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18478);