C₅×#9422 #11932142

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9422

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

384 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 640

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

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

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, 2410);