C₅×#9514 #11932234

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9514

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 478 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,11,15,18)(9,12)(10,13)(14,17,20,21)(16,19)(22,23,26,30,27,24,28,32)(25,31,34,37,33,29,35,36);(6,8,10,15)(7,11)(9,14,16,20)(12,17)(13,18)(19,21)(22,24,26,32,27,23,28,30)(25,31,35,36,33,29,34,37);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,17,18,21)(22,25)(23,29)(24,31)(26,35)(27,33)(28,34)(30,37)(32,36)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2502);