C₃₇×#10624677 #41004466

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

Aliases

direct product

C₃₇×#10624677

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

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

normal subgroups only

This group has 198 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)(3,9,19)(4,6,14)(7,15,21)(8,18,12)(10,16,13)(11,23,26)(17,27,22)(20,25,24)(28,29,32)(30,36,47)(31,35,42)(33,43,39)(34,46,40)(37,44,51)(38,45,41)(48,52,50)(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);(2,6,8)(3,10,21)(7,16,9)(11,24,22)(13,15,19)(25,27,26)(28,30,37,40,38,50,31,39,49)(29,33,44,34,36,48,35,45,53)(32,41,51,46,43,52,42,47,54);(1,3,11,4,10,22,12,21,24)(2,7,17,6,9,20,8,16,23)(5,13,25,14,15,26,18,19,27)(29,34,35)(32,42,46)(33,36,45)(41,47,43)(44,48,53)(51,54,52)

Conjugacy classes & character table

1517 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 237);