C₅×#7192 #11929912

order	1280
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7192

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂³	C₄×C₅×C₈
Frattini, cosocle	C₂²×M₁₆	C₂²×C₅
socle	C₂²×C₅	#324
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(10,11,13,19,15,21,29,37,16,18,27,34,14,20,28,36)(12,22,23,35,25,32,39,40,26,17,30,31,24,33,38,41);(6,7,8,9)(10,12,14,24,16,26,15,25)(11,17,20,32,18,22,21,33)(13,23,28,38,27,30,29,39)(19,35,36,41,34,31,37,40)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 180);