C₉×C₃₇×#731 #41004506

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₉×C₃₇×#731

Subgroups

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

292 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 73
transitive degree	μ_t(G) = 8991

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,4,3,5,7,6,8,9)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(10,11,14)(12,18,22)(13,17,23)(15,24,20)(16,27,21)(19,29,32)(25,35,36)(26,34,31)(28,33,30);(10,12,19,13,20,30,21,31,36)(11,15,25,17,26,29,16,18,28)(14,22,32,23,24,33,27,34,35)

Conjugacy classes & character table

5661 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 277);