C₅×#20773 #11943493

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#20773

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

830 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 382 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) = 640

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 13761);