C₅×#9334 #11932054

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9334

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

398 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 114 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)(6,7,10,14,11,13,20,8)(9,16,18,21,17,12,19,15)(22,23,27,31,28,33,26,32)(24,34,37,44,38,47,36,29)(25,39,41,30,42,45,40,46)(35,43,50,52,51,53,49,48);(6,8,10,13,11,14,20,7)(9,15,19,16,17,21,18,12)(22,24)(23,29,33,44)(25,35)(26,36)(27,37)(28,38)(30,43,46,53)(31,34,32,47)(39,48,45,52)(40,49)(41,50)(42,51);(6,9,11,17)(7,12,13,16)(8,15,14,21)(10,19,20,18)(22,25,28,42)(23,30,33,46)(24,35,38,51)(26,41,27,40)(29,43,44,53)(31,39,32,45)(34,48,47,52)(36,50,37,49)

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, 2322);