C₅×#10612 #11933332

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10612

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 328 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) = 37
transitive degree	μ_t(G) = 320

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

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

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, 3600);