C₅×#9494 #11932214

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9494

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 514 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) = 160

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

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

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, 2482);