C₅×#9970 #11932690

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#9970

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₅×#2656	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

772 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 33
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2958);