C₃₇×#10624688 #41004477

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

Aliases

direct product

C₃₇×#10624688

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	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

434 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 2997

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

(1,2,4,3,5,7,6,8,9)(10,11,14)(12,16,21)(13,15,22)(17,26,20)(18,24,29)(19,23,30)(25,33,28)(27,31,35)(32,36,34)(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);(11,15,17)(16,23,25)(24,31,32);(10,12,18,13,19,27,20,28,34)(11,16,24,15,23,31,17,25,32)(14,21,29,22,30,35,26,33,36)

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, 248);