C₅×#12277 #11934997

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12277

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ G

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23)(24,28)(25,35)(26,39)(27,31)(29,38)(30,40)(32,47)(33,46)(34,43)(36,51)(37,50)(41,48)(42,49)(44,52)(45,53);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,24,27,34)(23,28,31,43)(25,36,37,32)(26,33,40,49)(29,44,45,41)(30,46,39,42)(35,47,50,51)(38,52,53,48);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,25)(23,29,31,45)(24,32)(26,38)(27,37)(28,41,43,44)(30,35,39,50)(33,48)(34,36)(40,53)(42,47,46,51)(49,52)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5265);