C₇₃×#12065 #32921469

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#12065

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#442
commutator	C₈	C₂²×C₈×C₇₃
Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₂×D₈×C₇₃	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ #322×C₇₃ ◁ G

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

normal subgroups only

This group has 230 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) = 97
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 97 points:

(1,2,4,7,5,3,6,8)(9,10,12,18,14,19,13,16)(11,20,21,24,22,15,23,17)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23);(1,3,4,8,5,2,6,7)(10,16,19,18)(11,22)(15,17,20,24)(21,23)

Conjugacy classes & character table

3358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5053);