C₇₃×#10665 #32920069

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

Aliases

direct product

C₇₃×#10665

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

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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:

(1,2,3,6,5,8,4,7)(9,10,13,18,14,19,15,29)(11,20,22,32,24,36,23,16)(12,25,27,35,28,17,30,34)(21,37,26,40,38,33,39,31)(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)(3,7)(4,6)(5,8)(9,11,15,22,14,24,13,23)(10,16,29,20,19,32,18,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(2,6)(3,4)(7,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,30)(16,31)(18,34)(19,25)(20,37)(22,26)(23,39)(24,38)(29,35)(32,40)(33,36)

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, 3653);