C₅×#9779 #11932499

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9779

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#390
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

454 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 640

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

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

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, 2767);