C₂×C₃×D₄×H(3,𝔽₃) #13048976

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂³×C₃⁴

Aliases

direct product	C₂×C₃×D₄×H(3,𝔽₃)
unambiguous semidirect product	C₄⋊(C₂²×C₃×H(3,𝔽₃))
unambiguous semidirect product	(C₂×C₃²×D₄)⋊C₃²

	subgroup	G/H
center, socle	C₂²×C₃²	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃³
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

1512 conjugacy classes of subgroups; only the 608 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 608 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(7,8,9);(3,4,5,6)(10,11,13)(12,15,17)(14,18,16);(4,6)(11,14,15)(13,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2694);