C₅×#12108 #11934828

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

Aliases

direct product

C₅×#12108

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#408 ◁ G

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

normal subgroups only

This group has 442 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) = 320

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

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

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, 5096);