C₅×#8797 #11931517

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#8797

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

228 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 640

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

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

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, 1785);