C₅×#2724 #10616319

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

Aliases

direct product

C₅×#2724

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

504 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11988009, #11988092, #11988212, …
maximal subgroup of	C₅×#7736, C₅×#7751, C₅×#7770, …
maximal normal subgroup of	C₅×#7736, C₅×#7751, C₅×#7770, …
maximal quotient of	C₅×#14384, C₅×#14440, C₅×#14446, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18401);