C₅×#12026 #11934746

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12026

Subgroups

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	#1568
prime radical	C₂³×C₅	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#97 ◁ C₅×#405 ◁ G

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,11,18,13,19,36,48,14,20,35,17,12,39,10,34)(8,21,24,41,26,50,23,42,27,46,37,63,25,59,57,15)(9,28,30,61,32,45,29,47,33,62,38,64,31,16,43,44)(22,51,53,40,55,65,67,60,56,69,58,66,54,49,52,68);(6,8,10,23,12,25,35,24,14,27,36,57,13,26,11,37)(7,15,17,21,19,41,34,50,20,42,18,46,39,63,48,59)(9,22,29,52,31,54,30,58,33,56,43,67,32,55,38,53)(16,40,44,49,28,60,61,51,45,66,47,65,62,68,64,69);(6,9,12,31,14,33,13,32)(7,16,19,28,20,45,39,62)(8,22,25,54,27,56,26,55)(10,29,35,30,36,43,11,38)(15,40,41,60,42,66,63,68)(17,44,34,61,18,47,48,64)(21,49,50,51,46,65,59,69)(23,52,24,58,57,67,37,53)

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, 5014);