C₅×#10217 #11932937

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10217

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,15)(9,11)(10,20)(12,14)(13,16)(17,21)(18,19)(22,23,28,33)(24,34,37,48)(25,38,41,46)(26,32,27,31)(29,42,44,36)(30,39,47,40)(35,43,51,45)(49,52,53,50);(6,8,11,17)(7,13,9,19)(10,18,12,16)(14,21,20,15)(22,24,28,37)(23,29,33,44)(25,35,41,51)(26,42,27,36)(30,45,47,43)(31,48,32,34)(38,49,46,53)(39,52,40,50);(6,9,11,7)(8,15,17,21)(10,20,12,14)(13,16,19,18)(22,25,28,41)(23,30,33,47)(24,35,37,51)(26,40,27,39)(29,43,44,45)(31,38,32,46)(34,49,48,53)(36,52,42,50)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3205);