C₅×#12163 #11934883

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12163

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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)(10,13)(14,15,19,23)(16,24,26,42)(17,28,30,44)(18,33,31,22)(20,35,36,25)(21,38,40,32)(27,43,34,39)(29,41,45,37);(6,7)(8,10)(9,11)(12,13)(14,16)(15,20)(17,29)(18,34)(19,26)(21,39)(22,41)(23,36)(24,28)(25,32)(27,31)(30,45)(33,37)(35,38)(40,43)(42,44);(6,8,9,12)(7,10,11,13)(17,30)(21,40)(24,42)(25,35)(28,44)(29,45)(32,38)(39,43)

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, 5151);