C₅₃×#10624658 #50311543

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

Aliases

direct product

C₅₃×#10624658

Subgroups

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₅₃×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

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

normal subgroups only

This group has 278 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) = 4293

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

(1,2,5,13,7,19,4,8,15)(3,9,14,11,18,26,12,6,16)(10,21,24,23,17,27,22,20,25)(28,29,32)(30,36,47)(31,35,42)(33,43,39)(34,46,40)(37,44,51)(38,45,41)(48,52,50)(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);(1,3,10)(2,6,17)(4,12,22)(5,14,24)(7,9,20)(8,18,21)(11,23,13)(15,16,25)(19,26,27)(28,30,37)(29,33,44)(31,39,49)(32,41,51)(34,36,48)(35,45,53)(38,50,40)(42,47,54)(43,52,46);(2,7,8)(3,11,12)(5,15,19)(6,18,9)(10,22,23)(24,27,25)(29,34,35)(32,42,46)(33,36,45)(41,47,43)(44,48,53)(51,54,52)

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, 218);