C₃₇×#10624929 #41004718

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃₇×#10624929

Subgroups

	subgroup	G/H
center	C₉×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

960 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 470 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) = 118
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 4 generators on 118 points:

(1,2,6)(3,11,19)(4,8,20)(5,10,21)(7,22,15)(9,29,18)(12,32,41)(13,25,42)(14,31,43)(16,26,44)(17,28,45)(23,48,35)(24,47,38)(27,54,40)(30,46,33)(34,57,64)(36,51,65)(37,56,66)(39,53,67)(49,70,60)(50,69,62)(52,73,63)(55,68,58)(59,75,78)(61,72,79)(71,81,77)(74,80,76)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(1,3,12)(2,7,23)(4,14,34)(5,15,33)(6,19,41)(8,24,49)(9,11,30)(10,25,32)(13,35,18)(16,37,59)(17,38,58)(20,43,64)(21,22,46)(26,50,71)(27,31,55)(28,51,57)(29,42,48)(36,60,40)(39,62,76)(44,66,78)(45,47,68)(52,56,74)(53,72,75)(54,65,70)(61,77,63)(67,69,80)(73,79,81);(3,13,15)(7,11,25)(12,33,35)(14,36,38)(19,42,22)(23,32,30)(24,31,51)(34,58,60)(37,61,62)(41,46,48)(43,65,47)(49,57,55)(50,56,72)(59,76,77)(64,68,70)(66,79,69)(71,75,74)(78,80,81);(1,4,16,5,17,39,18,40,63)(2,8,26,10,28,53,9,27,52)(3,14,37,15,38,62,13,36,61)(6,20,44,21,45,67,29,54,73)(7,24,50,25,51,72,11,31,56)(12,34,59,33,58,76,35,60,77)(19,43,66,22,47,69,42,65,79)(23,49,71,32,57,75,30,55,74)(41,64,78,46,68,80,48,70,81)

Conjugacy classes & character table

3885 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 489);