C₅×#12167 #11934887

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12167

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424 ◁ G

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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)(10,13)(14,15,20,26)(16,27,30,42)(17,21,34,38)(18,25,19,24)(22,35,40,29)(23,33,37,31)(28,41,32,39)(36,44,43,45);(6,7)(8,10)(9,11)(12,13)(15,21)(17,31)(18,19)(22,36)(23,25)(24,37)(26,38)(27,41)(28,44)(29,35)(32,45)(33,34)(39,42)(40,43);(6,8,9,12)(7,10,11,13)(14,16,20,30)(15,22,26,40)(17,32,34,28)(18,35,19,29)(21,36,38,43)(23,41,37,39)(24,42,25,27)(31,45,33,44)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5155);