C₅×#11917 #11934637

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11917

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1533
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#97	C₂³×C₅
prime radical	C₅×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×M₁₆ ◁ G

280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 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)(6,7,10,14,9,15,22,33,11,16,26,34,23,13,25,31)(8,17,19,35,20,12,27,29,21,28,36,32,18,30,24,37);(6,8)(7,12)(9,18)(10,24)(11,21)(13,28)(14,32)(15,17)(16,30)(19,25)(20,23)(22,36)(26,27)(29,33)(31,37)(34,35);(6,9,11,23)(7,13,16,15)(8,18,21,20)(10,25,26,22)(12,28,30,17)(14,33,34,31)(19,27,36,24)(29,35,37,32)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4905);