C₅×#10110 #11932830

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10110

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

498 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,15)(9,20)(10,13)(11,14)(12,17)(16,19)(18,21)(24,30)(25,32)(27,35)(28,33)(31,34)(36,37);(6,8,11,19)(7,12,14,21)(9,18,10,17)(13,16,20,15)(22,23,26,29)(24,27,31,36)(25,33,32,28)(30,37,34,35);(6,7)(8,16)(9,13)(10,20)(11,14)(12,17)(15,19)(18,21)(22,24)(23,27)(25,34)(26,31)(28,37)(29,36)(30,32)(33,35)

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, 3098);