C₇₃×#9779 #32919183

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₃×#9779

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#390
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

454 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 9344

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

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

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, 2767);