C₅×#32705 #11955425

order	1280
nilpotency class	3
isoclinism	isoclinic class of #31077
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#32705

Subgroups

	subgroup	G/H
center	C₄×C₅	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	#2804
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

536 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25693);