C₂₇×#328 #13134071

order	1728
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂₇×#328

Subgroups

	subgroup	G/H
center	C₂²×C₂₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₂₇
Frattini, cosocle	C₂²×C₄×C₉	C₂²×C₃
socle	C₂²×C₃	C₉×C₂²⋊C₄
prime radical	C₂³×C₃	D₄×C₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ C₂²×C₄×C₂₇ ◁ G

160 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 432

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

513 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 134);