C₅×#13354 #11936074

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13354

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3021	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

692 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 94 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)(7,12)(8,16)(9,19)(10,23)(13,27)(14,30)(15,20)(17,34)(18,24)(21,33)(25,28)(26,32)(29,37)(31,35);(6,7)(8,13)(9,20)(10,14)(11,15)(12,19)(16,28)(17,35)(18,29)(21,25)(22,30)(23,32)(24,31)(26,36)(27,34)(33,37);(6,8,11,18)(7,13,15,29)(9,21,19,17)(10,24,23,16)(12,25,20,35)(14,31,32,28)(22,33,36,34)(26,37,30,27)

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, 6342);