C₂×D₄×#732 #13047499

order	1296
nilpotency class	3
isoclinism	isoclinic class of D₄×AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃²

Aliases

direct product	C₂×D₄×#732
unambiguous semidirect product	C₄⋊(C₂²×#732)

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂³×C₃²
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃²×D₄ ◁ G

432 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 33
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1217);