C₅×#3249 #10616844

order	640
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#3249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

348 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 188 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)(14,15,18,21)(16,22,24,29)(17,25,27,20)(19,26,28,23);(6,7)(8,11)(9,10)(12,13);(7,10)(11,13)(14,16,18,24)(15,19,21,28)(17,26,27,23)(20,29,25,22);(6,8,9,12)(7,11,10,13)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12000254, #12000259, #12001391, …
maximal subgroup of	C₅×#10030, C₅×#10031, C₅×#10195, …
maximal normal subgroup of	C₅×#10030, C₅×#10031, C₅×#10195, …
maximal quotient of	C₅×#21885, C₅×#21961, C₅×#22113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18926);