C₅×#8796 #11931516

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×#8796

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

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, 1784);