C₅×#8794 #11931514

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅×#8794

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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)(6,7,11,8)(9,10,13,12)(14,15)(16,24,27,20)(17,28,31,43)(18,23)(19,22)(21,30,36,40)(25,39)(26,38,42,34)(29,44)(32,37)(33,41)(35,45);(6,8,10,13,11,7,12,9)(14,16,18,26,19,27,32,42)(15,20,23,34,22,24,37,38)(17,29,30,45,31,25,40,41)(21,35,28,39,36,33,43,44);(6,9,10,8,11,13,12,7)(14,17)(15,21,22,36)(16,25)(18,30)(19,31)(20,33,24,35)(23,28,37,43)(26,41)(27,29)(32,40)(34,44,38,39)(42,45)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1782);