C₇₃×#12364 #32921768

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#12364

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	C₇₃×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄×C₈×C₇₃ ◁ G

358 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

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, 5352);