C₃³×#359 #13146152

order	1728
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₃³×#359

Subgroups

	subgroup	G/H
center	C₃³×C₄	D₈
commutator	C₈	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₈	C₂²×C₃³
socle	C₂×C₃³	#102
prime radical	C₃³×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃³×C₄ ◁ C₂×C₃³×C₄ ◁ C₂×C₃³×C₈ ◁ G

784 conjugacy classes of subgroups; only the 420 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 420 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) = 41
transitive degree	μ_t(G) = 864

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

594 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 12215);