C₅₃×#10624702 #50311587

order	38637
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₅₃ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₅₃×#10624702

Subgroups

	subgroup	G/H
center	C₉×C₅₃	C₃×H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₅₃
Frattini, cosocle	C₃×C₉	C₃³×C₅₃
socle	C₃×C₅₃	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

312 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Conjugacy classes & character table

5565 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 262);