C₅×#11871 #11934591

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11871

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1519
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

360 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,11,17,13,18,12,37)(8,19,22,45,24,50,23,58)(9,25,28,44,30,60,29,15)(10,31,34,16,36,47,35,48)(14,32,39,57,41,56,40,21)(20,51,26,61,54,68,53,49)(27,46,63,59,33,43,64,65)(38,52,66,69,67,55,42,62);(6,8,13,24)(7,14,18,41)(9,26,30,53)(10,32,36,56)(11,23,12,22)(15,42,44,66)(16,45,48,58)(17,39,37,40)(19,47,50,31)(20,29,54,28)(21,35,57,34)(25,38,60,67)(27,52,33,55)(43,49,46,61)(51,59,68,65)(62,64,69,63);(6,9,12,29,13,30,11,28)(7,15,17,25,18,44,37,60)(8,20,23,53,24,54,22,26)(10,33,34,64,36,27,35,63)(14,38,40,42,41,67,39,66)(16,46,31,65,48,43,47,59)(19,49,45,51,50,61,58,68)(21,55,32,62,57,52,56,69)

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, 4859);