C₅₃×#10624626 #50311511

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

Aliases

direct product

C₅₃×#10624626

Subgroups

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

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

normal subgroups only

This group has 186 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:

(1,2,6,15,9,24,5,10,18)(3,8,16,14,22,17,4,7,19)(11,23,25,13,20,26,12,21,27)(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);(1,3,11,5,4,12,15,14,13)(2,7,20,10,22,23,9,8,21)(6,16,25,18,19,27,24,17,26)(28,30,37,31,38,49,32,39,48)(29,34,44,36,46,47,35,45,53)(33,40,50,41,51,54,42,43,52);(1,4,13,5,14,11,15,3,12)(2,8,23,10,7,21,9,22,20)(6,17,27,18,16,26,24,19,25)(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, 186);