C₃₇×#10624936 #41004725

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

Aliases

direct product

C₃₇×#10624936

Subgroups

	subgroup	G/H
center, socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃×3₋¹⁺²×C₃₇ ◁ G

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

normal subgroups only

This group has 948 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)(7,22,14)(9,29,18)(12,32,41)(13,31,42)(15,26,43)(16,27,44)(17,28,45)(23,49,61)(24,47,37)(25,48,38)(30,46,34)(33,57,64)(35,56,65)(36,52,66)(39,53,67)(40,54,68)(50,71,59)(51,70,62)(55,69,60)(58,75,78)(63,73,79)(72,81,77)(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);(1,3,12,18,38,61,5,14,34)(2,7,23,9,11,30,10,25,32)(4,13,33,16,36,59,17,37,60)(6,19,41,29,48,49,21,22,46)(8,24,50,27,31,55,28,52,57)(15,35,58,40,63,77,39,62,76)(20,42,64,44,66,71,45,47,69)(26,51,72,54,56,74,53,73,75)(43,65,78,68,79,81,67,70,80);(1,4,15,5,17,39,18,16,40)(2,8,26,10,28,53,9,27,54)(3,13,35,14,37,62,38,36,63)(6,20,43,21,45,67,29,44,68)(7,24,51,25,52,73,11,31,56)(12,33,58,34,60,76,61,59,77)(19,42,65,22,47,70,48,66,79)(23,50,72,32,57,75,30,55,74)(41,64,78,46,69,80,49,71,81);(4,16,17)(8,27,28)(13,36,37)(15,39,40)(20,44,45)(24,31,52)(26,53,54)(33,59,60)(35,62,63)(42,66,47)(43,67,68)(50,55,57)(51,73,56)(58,76,77)(64,71,69)(65,70,79)(72,75,74)(78,80,81)

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, 496);