C₅×#13532 #11936252

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13532

Subgroups

	subgroup	G/H
center	C₄×C₅	#493
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#2093
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

312 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,7,10,16,21,31,11,17,12,19,27,37,13,18,29,36)(8,20,22,35,30,34,28,32,24,33,9,26,25,15,23,14);(6,8,21,25,12,24,13,30)(7,14,31,26,19,32,18,35)(9,27,28,29,22,10,23,11)(15,17,33,37,34,36,20,16);(6,9)(7,15)(8,10)(11,25)(12,22)(13,23)(14,17)(16,35)(18,20)(19,34)(21,28)(24,27)(26,37)(29,30)(31,33)(32,36)

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, 6520);