C₅×#8327 #11931047

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8327

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,32)(25,36,38,29)(26,39)(27,31)(28,34)(30,41)(33,47,49,46)(35,45)(37,44,53,51)(40,43)(42,48,50,52);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,27,35)(23,28,31,43)(25,33,38,49)(26,40,41,34)(29,42,36,50)(30,45,39,32)(37,52,53,48)(44,47,51,46);(8,14)(11,17)(15,20)(18,21)(22,25,27,38)(23,29,31,36)(24,33,35,49)(26,37,41,53)(28,42,43,50)(30,44,39,51)(32,46,45,47)(34,48,40,52)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1315);