C₅₃×#10624654 #50311539

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

Aliases

direct product

C₅₃×#10624654

Subgroups

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

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

normal subgroups only

This group has 224 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)(3,9,14)(4,8,15)(6,16,12)(7,19,13)(10,21,24)(11,18,25)(17,27,23)(20,26,22)(29,32,35)(34,40,44)(43,48,51)(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);(1,3,10,13,11,23,4,12,22)(2,6,17,7,9,20,8,18,21)(5,14,24,19,25,27,15,16,26)(28,29,33)(30,34,41)(31,35,42)(32,39,38)(36,43,49)(37,44,50)(40,47,46)(45,51,54)(48,53,52);(3,11,12)(6,9,18)(10,22,23)(14,25,16)(17,21,20)(24,26,27)(28,30,36,31,37,45,38,46,52)(29,34,43,35,44,51,32,40,48)(33,41,49,42,50,54,39,47,53)

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, 214);