C₅×#12865 #11935585

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12865

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1820
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×#41 ◁ G

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,17,20,15)(9,21)(10,14)(11,16)(12,19,27,34)(13,18,30,22)(23,26)(24,32)(25,31)(28,37)(29,33,35,36);(6,8)(7,12,16,27)(9,22)(10,19)(11,20)(13,28,30,26)(14,17,31,15)(18,24)(21,35,32,29)(23,36)(25,34)(33,37);(6,9,10,23,11,24,25,37)(7,13,14,29,16,30,31,35)(8,18,19,33,20,22,34,36)(12,26,17,32,27,28,15,21)

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, 5853);