C₅×#8754 #11931474

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#8754

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#381
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 566 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)(8,12,13,10)(9,11)(14,15,18,23,19,22,32,37)(16,24,26,38,27,20,31,34)(17,28,30,42,29,21,35,36)(25,33,41,44,40,39,43,45);(14,16,19,27)(15,20,22,24)(17,25,29,40)(18,31,32,26)(21,33,28,39)(23,38,37,34)(30,43,35,41)(36,45,42,44);(6,8,9,13)(7,10,11,12)(14,17)(15,21,22,28)(16,25)(18,30)(19,29)(20,33,24,39)(23,36,37,42)(26,41)(27,40)(31,43)(32,35)(34,44,38,45)

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, 1742);