C₇₃×#13540 #32922944

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₃×#13540

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂×C₇₃	#2099
prime radical	C₂³×C₇₃	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

234 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 105
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 105 points:

(1,2,5,7,6,11,8,12)(3,13,15,26,17,22,16,9)(4,18,20,28,21,10,24,23)(14,19,29,31,27,32,30,25)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,8,15,6,17,5,16)(2,9,12,13,11,26,7,22)(4,19,24,31,21,32,20,25)(10,27,28,30,18,14,23,29);(1,4,5,20,6,21,8,24)(2,10,7,23,11,18,12,28)(3,14,15,29,17,27,16,30)(9,25,13,19,26,31,22,32)

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, 6528);