C₅×#10387 #11933107

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10387

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

414 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 90 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)(7,12,14,17)(9,20)(13,29,21,25)(15,28,31,32)(18,35)(22,23)(27,30,34,37)(33,36);(7,13,14,21)(8,16)(12,25,17,29)(15,30,31,37)(18,33)(19,26)(27,32,34,28)(35,36);(6,7,11,15)(8,17,16,32)(9,21,23,37)(10,14,24,31)(12,26,28,19)(13,20,30,22)(18,27,33,25)(29,36,34,35)

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, 3375);