C₅×#12299 #11935019

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12299

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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)(6,7,10,24,11,15,26,14)(8,16,19,29,18,32,25,33)(9,20,23,13,22,36,37,30)(12,21,31,17,28,34,27,35);(6,8,11,18)(7,12,15,28)(9,21,22,34)(10,25,26,19)(13,29,30,33)(14,31,24,27)(16,20,32,36)(17,23,35,37);(13,30)(16,32)(20,36)(29,33)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5287);