C₃₇×#10624645 #41004434

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

Aliases

direct product

C₃₇×#10624645

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

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

normal subgroups only

This group has 192 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)(3,7,12)(4,6,13)(8,17,11)(9,14,21)(10,16,22)(15,24,18)(19,23,26)(20,25,27)(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)(3,9,18)(7,15,14)(10,20,19)(12,21,24)(22,27,26)(28,30,36)(29,33,43)(31,38,48)(32,40,50)(34,44,53)(35,47,45)(37,49,39)(41,46,54)(42,52,51);(1,3,10,4,9,19,11,18,20)(2,7,16,6,14,23,8,15,25)(5,12,22,13,21,26,17,24,27)

Conjugacy classes & character table

2109 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 205);