C₅×#12119 #11934839

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12119

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

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

normal subgroups only

This group has 404 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)(8,19)(9,13)(10,17)(11,16)(12,18)(14,21)(15,23)(20,32)(22,33)(24,29)(25,31)(26,28)(27,36)(30,37)(34,35);(7,13)(9,23)(10,11)(14,27)(15,17)(16,28)(18,29)(19,32)(20,35)(21,25)(24,26)(30,33)(31,34)(36,37);(6,8,12,22)(7,14,18,31)(9,20,24,37)(10,25,11,21)(13,27,29,34)(15,32,28,30)(16,33,17,19)(23,35,26,36)

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, 5107);