C₅×#7480 #11930200

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7480

Subgroups

	subgroup	G/H
center	C₄×C₅	#357
commutator	C₁₆	C₄²×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
socle	C₂×C₅	#1380
prime radical	C₂³×C₅	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 166 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

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

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, 468);