#41×#737 #13048986

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product

#41×#737

Subgroups

	subgroup	G/H
center	C₄×C₉	C₂²×C₃²
commutator, Frattini, socle, cosocle	C₂×C₃	C₂³×C₃³
prime radical	#41×H(3,𝔽₃)	C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ G

660 conjugacy classes of subgroups; only the 493 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 493 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) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2704);