C₅×#10227 #11932947

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10227

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

466 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 122 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)(7,11)(8,12)(9,13)(14,15,20,25)(16,26,29,37)(17,22,33,40)(18,24,19,23)(21,30,36,28)(27,42,43,38)(31,45,44,35)(32,41,34,39);(6,7,10,8)(9,12,13,11)(14,16,20,29)(15,21,25,36)(17,31,33,44)(18,30,19,28)(22,38,40,42)(23,37,24,26)(27,34,43,32)(35,41,45,39);(6,8,10,7)(9,11,13,12)(14,17,20,33)(15,22,25,40)(16,27,29,43)(18,32,19,34)(21,35,36,45)(23,39,24,41)(26,38,37,42)(28,31,30,44)

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, 3215);