C₃×C₅×#6872 #16814210

order	3645
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅×C₉ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃×C₅×#6872

Subgroups

	subgroup	G/H
center	C₃×C₅×C₉	H(3,𝔽₃)
commutator	C₃²	C₃²×C₅×C₉
Frattini, cosocle	C₃×C₉	C₃³×C₅
socle	C₃²×C₅	C₃²⋊C₉
prime radical	C₃×C₅×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₉ ◁ C₃²×C₅×C₉ ◁ G

320 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 89
transitive degree	μ_t(G) = 1215

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,3)(85,86,87,88,89);(4,5,9,7,12,24,19,30,46,8,13,23,20,31,45,41,55,68,21,11,27,42,29,51,65,54,72)(6,14,32,17,34,57,39,59,76,18,28,50,40,53,71,64,74,82,16,10,22,38,26,44,63,49,67)(15,33,43,36,58,66,61,77,80,35,52,56,60,73,75,78,83,84,37,25,47,62,48,69,79,70,81);(4,6,15)(5,10,25)(7,17,36)(8,18,35)(9,22,43)(11,28,52)(12,26,48)(13,14,33)(16,37,21)(19,39,61)(20,40,60)(23,32,56)(24,44,66)(27,50,47)(29,53,73)(30,49,70)(31,34,58)(38,62,42)(41,64,78)(45,57,75)(46,67,80)(51,71,69)(54,74,83)(55,59,77)(63,79,65)(68,76,84)(72,82,81)

Conjugacy classes & character table

765 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3645, 261);