C₅×#12240 #11934960

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12240

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ G

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

normal subgroups only

This group has 350 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) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23)(24,27)(25,28)(26,30)(29,34)(31,36)(32,37)(33,35);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,24,26,31)(23,27,30,36)(25,33,34,32)(28,37,29,35);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(23,28)(24,31)(25,34)(27,35)(29,30)(36,37)

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, 5228);