C₅×#2834 #10616429

order	640
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#2834

Subgroups

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

372 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 168 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)(11,13)(14,15)(16,18)(17,19)(20,21)(22,23,26,28)(24,25,27,29);(8,12)(11,13)(14,15,17,19)(16,20,21,18)(22,24)(23,25)(26,27)(28,29);(6,7,9,10)(8,11,12,13)(14,15)(16,20)(17,19)(18,21)(22,25)(23,27)(24,28)(26,29);(6,8)(7,11)(9,12)(10,13)(14,16)(15,18)(17,21)(19,20)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12011610, #12012196, D₅×#2834, …
maximal subgroup of	C₅×#12138, C₅×#12232, C₂×C₅×#2834, …
maximal normal subgroup of	C₅×#12138, C₅×#12232, C₂×C₅×#2834, …
maximal quotient of	C₅×#14627, C₅×#14636, C₅×#14644, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18511);