C₇₃×#10227 #32919631

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

Aliases

direct product

C₇₃×#10227

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

466 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 3215);