C₇₃×#11669 #32921073

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#11669

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#418
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₄×C₈	C₂³×C₇₃
socle	C₂×C₇₃	#1490
prime radical	#575×C₇₃	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ C₄×M₁₆×C₇₃ ◁ G

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

normal subgroups only

This group has 360 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) = 89
transitive degree	μ_t(G) = 1168

A minimal faithful permutation representation — 3 generators on 89 points:

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

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, 4657);