C₅×#10081 #11932801

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#10081

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#2819	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

648 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 640

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

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

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, 3069);