C₅×#12864 #11935584

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12864

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1820
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×#41 ◁ G

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,17,20,15)(9,21)(10,14)(11,16)(12,19,26,33)(13,18,28,32)(22,25)(23,30)(24,29)(27,31,36,37)(34,35);(6,8)(7,12,16,26)(9,18)(10,19)(11,20)(13,25,28,35)(14,17,29,15)(21,27,30,36)(22,31)(23,32)(24,33)(34,37);(6,9,10,22,11,23,24,34)(7,13,14,27,16,28,29,36)(8,18,19,31,20,32,33,37)(12,25,17,30,26,35,15,21)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5852);