C₇₃×#8534 #32917938

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇₃×#8534

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#379
commutator	C₂⁴	C₂²×C₄×C₇₃
Frattini, cosocle	C₂⁵	C₂³×C₇₃
prime radical	C₇₃×#3484	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂⁵×C₇₃ ◁ G

1408 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 158 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:

(2,6,8,13)(3,9)(7,12,14,16)(10,15)(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);(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16);(3,10)(6,12)(9,15)(13,16)

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1522);