C₅×#9190 #11931910

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9190

Subgroups

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

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

normal subgroups only

This group has 436 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)(7,10,11,12)(14,15,19,23,20,24,18,34)(16,25,29,36,28,40,27,21)(17,30,32,38,33,22,31,37)(26,39,43,35,42,45,41,44);(6,7,8,12,9,11,13,10)(14,16,20,28)(15,21)(17,26,33,42)(18,27,19,29)(22,35)(23,25)(24,36)(30,44)(31,41,32,43)(34,40)(37,45)(38,39);(14,17,20,33)(15,22,24,30)(16,26,28,42)(18,32,19,31)(21,35,36,44)(23,38,34,37)(25,39,40,45)(27,43,29,41)

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, 2178);