C₂×H(3,𝔽₃)×D₁₆ #13145274

order	1728
nilpotency class	4
isoclinism	isoclinic class of H(3,𝔽₃)×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃²

Aliases

direct product	C₂×H(3,𝔽₃)×D₁₆
unambiguous semidirect product	(C₂×C₃²×D₁₆)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×D₈
commutator, Frattini, cosocle	C₃×C₈	C₂³×C₃²
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄×H(3,𝔽₃) ◁ C₂×C₈×H(3,𝔽₃) ◁ G

539 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11337);