C₅₃×#10624481 #50311366

order	38637
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₅₃×#10624481

Subgroups

	subgroup	G/H
center, socle	C₃²×C₅₃	#732
commutator, Frattini, cosocle	C₃²×C₉	C₃²×C₅₃
prime radical	C₃×C₅₃×#732	C₃
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

144 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 89
transitive degree	μ_t(G) = 4293

A minimal faithful permutation representation — 2 generators on 89 points:

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

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, 41);