C₅₃×#10624854 #50311739

order	38637
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₃³×C₅₃ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₅₃×#10624854

Subgroups

	subgroup	G/H
center, socle	C₃×C₅₃	#6911
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₅₃
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ C₃²×C₅₃ ◁ C₃×3₋¹⁺²×C₅₃ ◁ G

178 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Conjugacy classes & character table

2173 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 414);