C₅₃×#10624756 #50311641

order	38637
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₅₃ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₅₃×#10624756

Subgroups

	subgroup	G/H
center, socle	C₃×C₅₃	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ C₃³×C₅₃ ◁ G

414 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 134
transitive degree	μ_t(G) = 4293

A minimal faithful permutation representation — 3 generators on 134 points:

(1,2,6)(3,11,30)(4,16,21)(5,10,22)(7,20,14)(8,26,18)(9,29,19)(12,31,44)(13,24,50)(15,25,52)(17,41,46)(23,47,38)(27,55,67)(28,54,43)(32,58,37)(33,48,68)(34,49,56)(35,62,73)(36,39,45)(40,53,42)(51,74,64)(57,72,61)(59,69,63)(60,76,79)(65,66,70)(71,81,77)(75,80,78)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134);(1,3,12)(2,7,23)(4,13,33)(5,15,34)(6,20,44)(8,24,48)(9,25,49)(10,11,31)(14,38,19)(16,39,66)(17,35,60)(18,37,61)(21,45,68)(22,30,56)(26,50,72)(27,51,71)(28,32,57)(29,52,47)(36,65,43)(40,62,76)(41,59,75)(42,64,77)(46,69,79)(53,73,81)(54,58,70)(55,74,80)(63,78,67);(1,4,17,5,18,42,19,43,67)(2,8,27,10,28,41,9,16,40)(3,13,35,15,37,64,14,36,63)(6,21,46,22,26,53,29,54,55)(7,24,51,11,32,59,25,39,62)(12,33,60,34,61,77,38,65,78)(20,45,69,30,50,73,52,58,74)(23,48,71,31,57,75,49,66,76)(44,68,79,56,72,81,47,70,80)

Conjugacy classes & character table

3021 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 316);