C₇₃×#10667 #32920071

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

Aliases

direct product

C₇₃×#10667

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂²×C₇₃	#399
prime radical	#532×C₇₃	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3655);