C₅×#9429 #11932149

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,11,13,10,8,9,12)(14,15,19,23,20,25,18,24)(16,26,29,36,30,39,28,21)(17,31,33,22,34,37,32,38)(27,35,42,44,43,45,41,40);(6,8,10,7)(9,12,11,13)(14,16)(15,21,25,36)(17,27)(18,28)(19,29)(20,30)(22,35,38,45)(23,26,24,39)(31,40,37,44)(32,41)(33,42)(34,43);(9,11)(12,13)(14,17,20,34)(15,22,25,38)(16,27,30,43)(18,33,19,32)(21,35,36,45)(23,31,24,37)(26,40,39,44)(28,42,29,41)

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, 2417);