C₅₃×#10624758 #50311643

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

Aliases

direct product

C₅₃×#10624758

Subgroups

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

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

normal subgroups only

This group has 402 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:

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

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, 318);