C₂×3₋¹⁺²×Q₃₂ #13145281

order	1728
nilpotency class	4
isoclinism	isoclinic class of H(3,𝔽₃)×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×3₋¹⁺²×Q₃₂
unambiguous semidirect product	(C₂×C₉×Q₃₂)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×D₈
commutator, Frattini, cosocle	C₃×C₈	C₂³×C₃²
prime radical	C₂²×C₃²	C₃×D₈
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ C₂×C₈×3₋¹⁺² ◁ G

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

normal subgroups only

This group has 328 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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2)(35,36,38,37,40,42,41,39,43);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20)(36,39,40)(38,42,43);(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11344);