C₅×#13423 #11936143

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13423

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄ ◁ G

558 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6411);