C₅×#11951 #11934671

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11951

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

340 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 640

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

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4939);