C₃₇×#10624853 #41004642

order	26973
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃₇×#10624853

Subgroups

	subgroup	G/H
center, socle	C₃×C₃₇	#6911
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₃₇
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃²×C₃₇ ◁ C₃×3₋¹⁺²×C₃₇ ◁ G

190 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 118 points:

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

Conjugacy classes & character table

1517 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 413);