C₇₃×#10643 #32920047

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#10643

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

298 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 113
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 113 points:

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

Conjugacy classes & character table

2701 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3631);