C₇₃×#10528 #32919932

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

Aliases

direct product

C₇₃×#10528

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

342 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,6)(4,7)(5,8)(9,10,15,19)(11,20,24,35)(12,25,28,17)(13,30,14,18)(16,23,32,22)(21,36,38,33)(26,31,37,40)(27,34,29,39)(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)(9,11,15,24)(10,16,19,32)(12,26,28,37)(13,23,14,22)(17,33,25,36)(18,35,30,20)(21,27,38,29)(31,39,40,34);(2,6)(3,4)(7,8)(9,12,13,27,15,28,14,29)(10,17,18,34,19,25,30,39)(11,21,22,37,24,38,23,26)(16,31,20,33,32,40,35,36)

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3516);