C₅×#11484 #11934204

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11484

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×#99
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	#1459
prime radical	C₅×#525	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

368 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,9,14,10,15,20,27,11,12,19,23,21,24,31,34)(8,13,17,25,18,26,30,35,16,22,28,32,29,33,36,37);(7,12)(8,16)(15,24)(17,28)(18,29)(25,32)(30,36)(35,37);(6,8,10,18,11,16,21,29)(7,13,15,26,12,22,24,33)(9,17,20,30,19,28,31,36)(14,25,27,35,23,32,34,37)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4472);