C₅₃×#10624624 #50311509

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

Aliases

direct product

C₅₃×#10624624

Subgroups

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

204 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 107
transitive degree	μ_t(G) = 12879

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2,5,12,8,18,4,6,14)(3,9,19,23,7,15,11,17,13)(10,21,25,27,20,24,22,16,26)(28,29,33,32,35,42,31,36,41)(30,34,43,39,45,51,38,46,40)(37,47,50,48,44,52,49,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(2,6,8)(7,9,17)(16,20,21)(28,30,37)(29,34,44)(31,38,49)(32,39,48)(33,40,50)(35,45,53)(36,46,47)(41,51,54)(42,43,52);(1,3,10,4,11,22,12,23,27)(2,7,16,6,9,20,8,17,21)(5,13,24,14,15,25,18,19,26)(29,35,36)(33,41,42)(34,45,46)(40,51,43)(44,53,47)(50,54,52)

Conjugacy classes & character table

3869 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 184);