SD₁₆×#731 #13046900

order	1296
nilpotency class	3
isoclinism	isoclinic class of D₈×AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

SD₁₆×#731

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	D₄×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²×C₄	C₂²×C₃²
prime radical	D₄×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×C₄ ◁ G

140 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 35 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 618);