C₅×#12764 #11935484

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12764

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄ ◁ G

768 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 41 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5752);