C₅×#12438 #11935158

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12438

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,17)(9,22)(10,23)(13,24)(15,29)(16,28)(18,19)(20,35)(21,30)(25,31)(26,27)(32,33)(36,37);(6,7,13,29,10,15,9,12,11,16,22,27,23,26,24,28)(8,14,30,36,19,32,18,25,21,34,17,31,20,33,35,37);(6,8)(7,14)(9,18)(10,19)(11,21)(12,25)(13,30)(15,32)(16,34)(17,22)(20,23)(24,35)(26,33)(27,31)(28,37)(29,36)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5426);