C₅×#12116 #11934836

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12116

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

426 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,17)(9,14)(10,22)(11,26)(12,16)(13,19)(15,24)(18,33)(20,34)(21,32)(23,28)(25,37)(27,31)(29,36)(30,35);(6,8)(7,13)(9,18)(10,23)(11,27)(12,20)(14,28)(15,31)(16,29)(17,22)(19,24)(21,36)(25,35)(26,30)(32,33)(34,37);(6,9,12,11)(7,14,16,26)(8,18,20,27)(10,24,25,21)(13,28,29,30)(15,22,32,37)(17,33,34,31)(19,35,36,23)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5104);