C₅×#2372 #10615967

order	640
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#2372

Subgroups

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

560 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 280 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)(6,7,9,8)(11,14)(15,17)(18,19,22,21)(20,24,25,23);(6,8,9,7)(19,21)(23,24);(10,11)(12,15)(13,14)(16,17)(18,20,22,25)(19,23,21,24);(10,12,13,16)(11,15,14,17)(18,21,22,19)(20,24,25,23)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11996919, #11996929, #12016099, …
maximal subgroup of	C₅×#9439, C₅×#9441, C₅×#12965, …
maximal normal subgroup of	C₅×#9439, C₅×#9441, C₅×#12965, …
maximal quotient of	C₅×#13805, C₅×#13816, C₅×#13824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18049);