C₅×#10216 #11932936

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 3204);