C₃₇×#10624788 #41004577

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

Aliases

direct product

C₃₇×#10624788

Subgroups

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

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, 348);