C₅×#12115 #11934835

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

Aliases

direct product

C₅×#12115

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#2010	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 422 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)(8,16)(9,20)(10,25)(11,15)(12,17)(13,24)(14,23)(18,36)(19,33)(21,31)(22,32)(26,29)(27,35)(28,37)(30,34);(6,8,11,19)(7,12,15,27)(9,21,22,37)(10,26,23,18)(13,29,30,36)(14,31,25,28)(16,32,33,20)(17,34,35,24);(9,22)(17,35)(21,37)(24,34)

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, 5103);