C₅₃×#10624764 #50311649

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

Aliases

direct product

C₅₃×#10624764

Subgroups

	subgroup	G/H
center, socle	C₃×C₅₃	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₅₃×#6909	C₃
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:

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

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, 324);