C₇₃×#10628 #32920032

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₃×#10628

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂²×C₇₃	#397
prime radical	C₂³×C₇₃	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

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

normal subgroups only

This group has 292 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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3616);