C₃₇×#10624621 #41004410

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

Aliases

direct product

C₃₇×#10624621

Subgroups

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

200 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 80 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)(7,15,23)(8,18,12)(10,21,25)(11,17,13)(16,26,22)(20,24,27)(28,29,32,39,45,51,31,34,41)(30,35,46,37,44,40,38,33,42)(36,43,50,49,47,52,48,53,54)(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)(7,9,17)(16,20,21)(28,30,36,31,38,48,39,37,49)(29,33,43,34,44,53,45,35,47)(32,40,50,41,46,54,51,42,52);(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)

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, 181);