C₅₃×#10624671 #50311556

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

Aliases

direct product

C₅₃×#10624671

Subgroups

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

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

normal subgroups only

This group has 236 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:

(1,2,5,4,8,15,13,7,19)(3,9,14,12,6,16,11,18,26)(10,21,24,22,20,25,23,17,27)(28,29,32,40,34,46,31,35,42)(30,36,47,38,45,41,39,33,43)(37,44,51,50,48,52,49,53,54)(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);(1,3,10,4,12,22,13,11,23)(2,6,17,8,18,21,7,9,20)(5,14,24,15,16,25,19,26,27)(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, 231);