C₅₃×#10624750 #50311635

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

Aliases

direct product

C₅₃×#10624750

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:

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

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, 310);