C₃₇×#10624623 #41004412

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₃₇×#10624623

Subgroups

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃₇×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

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

normal subgroups only

This group has 218 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) = 91
transitive degree	μ_t(G) = 8991

A minimal faithful permutation representation — 3 generators on 91 points:

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

Conjugacy classes & character table

2701 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 183);