C₅₃×#10624888 #50311773

order	38637
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₅₃ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₅₃×#10624888

Subgroups

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

764 conjugacy classes of subgroups; only the 440 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 440 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 — 4 generators on 107 points:

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

Conjugacy classes & character table

5565 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 448);